The Kester Function: A Function Defined by the Continuity of Its Underlying Function

THE KESTER FUNCTION

A Function Defined by the Continuity of Its Underlying Function

Kester Pembroke

ABSTRACT

We introduce the Kester function, an operator that modifies a real-valued function according to whether the function is continuous at each point.

Given a function f : X → ℝ, the Kester function K[f] is defined by:

Kf =
2f(x), if f is continuous at x,
f(x),  if f is discontinuous at x.

The construction is unusual because the value of K[f] at x depends not merely on f(x), but also on a local property of f at that point. We investigate the basic properties of this operator, its relationship with the continuity set of f, its iterates, fixed points, and possible generalizations.

1. INTRODUCTION

Continuity is normally regarded as a property of a function rather than as part of the numerical value of the function.

For a function f, one asks whether:

lim(x→a) f(x) = f(a).

The answer is either yes or no.

The Kester function uses this property of f as part of the rule that defines a new function. At a point where f is continuous, its value is doubled. At a point where f is discontinuous, its value is left unchanged.

Let C_f denote the set of points at which f is continuous:

C_f = {x : f is continuous at x}.

Let D_f denote the set of discontinuity points:

D_f = X \ C_f.

The Kester function is therefore:

Kf =
2f(x), if x ∈ C_f,
f(x),  if x ∈ D_f.

Equivalently,

Kf = (1 + 1_Cf(x))f(x),

where 1_Cf is the indicator function of the continuity set.

The central idea is therefore simple:

The function is transformed according to whether the function itself is continuous at the point being considered.

2. DEFINITION

Definition — Kester Function

Let f : X → ℝ be a function.

The Kester function associated with f is the function K[f] defined by:

Kf =
2f(x), if f is continuous at x,
f(x),  if f is discontinuous at x.

The factor 2 is not essential. More generally, for a constant a, we can define:

K_af =
af(x), if f is continuous at x,
f(x),  if f is discontinuous at x.

The original Kester function corresponds to a = 2.

3. BASIC EXAMPLES

3.1 A continuous function

Consider:

f(x) = x².

This function is continuous everywhere.

Therefore every point belongs to C_f, and hence:

Kf = 2x².

Thus, for a function that is continuous everywhere, the Kester function simply doubles the original function.

3.2 A function discontinuous at one point

Consider:

f(x) =
1, if x = 0,
0, if x ≠ 0.

This function is discontinuous at x = 0 and continuous everywhere else.

At x = 0, the Kester rule leaves the value unchanged because f is discontinuous there.

At every other point, f(x) = 0, so doubling it makes no difference.

Therefore:

Kf =
1, if x = 0,
0, if x ≠ 0.

In this example:

K[f] = f.

3.3 A jump discontinuity

Consider:

f(x) =
0, if x < 0,
1, if x ≥ 0.

The function is discontinuous at x = 0 and continuous everywhere else.

Therefore:

Kf =
0, if x < 0,
1, if x = 0,
2, if x > 0.

The Kester transformation therefore changes the size of the function on the continuous regions while leaving the value at the discontinuity itself unchanged.

4. THE CONTINUITY SET AS THE GOVERNING OBJECT

The Kester function can be written as:

K[f] = f + f·1_Cf.

Consequently:

K[f] − f = f·1_Cf.

This means that the difference between the Kester function and the original function is:

Kf − f(x) =
f(x), if f is continuous at x,
0,    if f is discontinuous at x.

Thus the transformation modifies the function precisely according to its continuity set.

This is the central structural feature of the construction.

Normally, a transformation of a function depends directly on its values. For example:

g(x) = f(x)²

depends only on f(x).

The Kester function is different. Its definition depends on both the value f(x) and the local behaviour of f around x.

5. WHAT HAPPENS TO CONTINUITY?

An important question is whether K[f] has the same continuity points as f.

The answer is not necessarily.

Suppose x₀ is inside the continuity set of f. In other words, there is a neighbourhood around x₀ in which f is continuous.

Within that neighbourhood:

Kf = 2f(x).

Since f is continuous there, K[f] is also continuous there.

Therefore:

If x₀ ∈ interior(C_f), then K[f] is continuous at x₀.

However, the boundary of the continuity set can behave differently.

At such a boundary, points where f is continuous may occur arbitrarily close to points where f is discontinuous.

The Kester multiplier can therefore change from 2 to 1 between arbitrarily nearby points.

This means that the Kester transformation can potentially introduce additional discontinuities.

The boundary of the continuity set is therefore a natural object to study.

6. ITERATION

The Kester function can be applied repeatedly.

Define:

f₀ = f,

f₁ = K[f],

f₂ = K[f₁],

and in general:

fₙ₊₁ = K[fₙ].

We can therefore study the sequence:

f, K[f], K²[f], K³[f], …

A simple case occurs when f is continuous everywhere.

Then:

K[f] = 2f.

Since 2f is also continuous everywhere:

K²[f] = 4f.

Continuing,

K³[f] = 8f,

and in general:

Kⁿ[f] = 2ⁿf.

Thus the Kester operator is not generally idempotent.

That is:

K[K[f]] ≠ K[f]

in general.

7. FIXED POINTS

A fixed point is a function satisfying:

K[f] = f.

We can determine exactly what such functions must look like.

Suppose x is a continuity point of f.

Then:

Kf = 2f(x).

For K[f] to equal f, we must have:

2f(x) = f(x).

Therefore:

f(x) = 0.

Hence, at every continuity point of a fixed point, the function must have value zero.

This gives the following result.

THEOREM

A function f satisfies:

K[f] = f

if and only if:

f(x) = 0

at every point where f is continuous.

PROOF

Suppose first that K[f] = f.

If x is a continuity point of f, then:

Kf = 2f(x).

Since Kf = f(x),

2f(x) = f(x),

so:

f(x) = 0.

Conversely, suppose that f(x) = 0 at every continuity point of f.

At a continuity point:

Kf = 2f(x) = 0 = f(x).

At a discontinuity point, the definition of the Kester function gives:

Kf = f(x).

Therefore:

K[f] = f.

8. THE SELF-REFERENTIAL ASPECT

One of the interesting features of the Kester function is its self-referential nature.

An ordinary pointwise transformation might look like:

g(x) = F(f(x)).

For example:

g(x) = f(x)².

The rule only needs to know the value f(x).

The Kester function instead asks a question about f itself:

“Is f continuous at x?”

The answer determines how f(x) is transformed.

Thus:

g(x) =
2f(x), if f is continuous at x,
f(x),  if f is discontinuous at x.

The function is therefore being modified according to one of its own mathematical properties.

This gives the construction a self-referential character, although the definition itself is mathematically straightforward and does not involve logical paradox.

9. GENERALIZATION

The factor 2 can be replaced by any real number a.

Define:

K_af =
af(x), if f is continuous at x,
f(x),  if f is discontinuous at x.

Some special cases are particularly simple.

If a = 1:

K₁[f] = f.

If a = 0:

K₀f =
0,    if f is continuous at x,
f(x), if f is discontinuous at x.

If a = 2:

K₂[f] = K[f],

which is the original Kester function.

This gives a whole family of continuity-dependent transformations rather than a single isolated construction.

10. POSSIBLE QUESTIONS FOR FURTHER RESEARCH

The definition raises several natural mathematical questions.

1. Which sets can occur as the continuity set of a function?
2. When does K[f] have exactly the same continuity set as f?
3. When does K[f] create new discontinuities?
4. Which functions are fixed points of K?
5. What happens when K is applied repeatedly?
6. Does the sequence
   f, K[f], K²[f], K³[f], …
   converge for interesting classes of functions?
7. How does the generalized operator K_a behave under repeated iteration?
8. Can the Kester construction be extended to functions between more general topological spaces?

These questions suggest that the definition can be viewed not simply as an isolated formula, but as the starting point for investigating a class of operators whose rules depend on regularity properties of the functions to which they are applied.

11. CONCLUSION

The Kester function is defined by:

Kf =
2f(x), if f is continuous at x,
f(x),  if f is discontinuous at x.

Its defining feature is that continuity, which is normally treated as a property of a function, becomes part of the rule used to construct another function.

The identity:

K[f] = f + f·1_Cf

shows that the construction is governed by the continuity set of the original function.

The operator has several elementary but interesting properties.

For continuous functions:

Kⁿ[f] = 2ⁿf.

Its fixed points are exactly the functions satisfying:

f(x) = 0

at every continuity point.

The generalized family K_a further shows that the construction naturally extends beyond the particular choice of the factor 2.

The Kester function therefore provides a simple example of a transformation in which a qualitative property of a function — continuity — directly determines a quantitative modification of that function.

Further investigation is required to determine the full mathematical significance and novelty of the construction.

AUTHOR’S NOTE

The term “Kester function” is used here as the proposed name for this construction. Before publication, a formal literature search should be carried out to determine whether an equivalent operator has already appeared under another name and to identify genuinely new theorems or properties that could form the basis of a research paper.

Absolutely. If the goal is just to put the idea on your Blogger now, I’d make it a readable mathematical blog post rather than a formal paper. Blogger can handle ordinary posts directly, and mathematical equations can be rendered with MathJax if you later want them displayed properly. 

Here is a version I think is much better for a first public post:

Introducing the Kester Function

By Kester Pembroke

I’ve been exploring a simple idea in mathematical analysis that I call the Kester function.

The idea is unusual because the value of a new function depends on whether the original function is continuous at that particular point.

The definition

Let f be a real-valued function.

Define its Kester function K[f] by

K[f](x)=
\begin{cases}
2f(x), & \text{if }f\text{ is continuous at }x,\\
f(x), & \text{if }f\text{ is discontinuous at }x.
\end{cases}

So the rule is very simple:

If f is continuous at x, double its value.

If f is discontinuous at x, leave its value alone.

The interesting part is that deciding what K[f](x) should be requires us to examine the behaviour of f around x, rather than just knowing the value f(x).

A simple example

Take

f(x)=x^2.

This function is continuous everywhere.

Therefore,

K[f](x)=2x^2.

Nothing surprising happens here.

But things become more interesting when a function has discontinuities.

Consider

f(x)=
\begin{cases}
0,&x<0,\\
1,&x\geq0.
\end{cases}

This function is discontinuous at x=0, but continuous everywhere else.

The Kester function becomes

K[f](x)=
\begin{cases}
0,&x<0,\\
1,&x=0,\\
2,&x>0.
\end{cases}

Notice what happened.

At the discontinuity itself, the value 1 was left unchanged.

Away from the discontinuity, the values were doubled.

The continuity set

Let

C_f=\{x:f\text{ is continuous at }x\}.

Then the Kester function can be written more compactly as

K[f](x)=
\left(1+\mathbf 1_{C_f}(x)\right)f(x),

where \mathbf 1_{C_f} is 1 on the continuity set and 0 outside it.

This gives

K[f]=f+f\mathbf1_{C_f}.

So the Kester function is essentially the original function plus an extra copy of f on its continuity set.

A surprising consequence

The really interesting question is not simply what the Kester function does to f.

It is:

What happens to the continuity of the new function?

At first it might seem that if f is continuous at a point, then K[f] should also be continuous there.

That is true in many cases—but not all.

Suppose x_0 is a continuity point of f, but it lies on the boundary between continuous and discontinuous points of f.

In other words, there are discontinuity points of f arbitrarily close to x_0.

At x_0,

K[f](x_0)=2f(x_0).

But along nearby discontinuity points,

K[f](x)=f(x).

Since f itself is continuous at x_0,

f(x)\to f(x_0).

Therefore, along those discontinuity points,

K[f](x)\to f(x_0),

while

K[f](x_0)=2f(x_0).

These can only agree if

f(x_0)=0.

This gives the following result.

Kester Boundary Theorem

Let C_f be the set of continuity points of f.

If

x_0\in C_f\cap\partial C_f,

then K[f] is continuous at x_0 if and only if

f(x_0)=0.

In other words:

A continuity point lying on the boundary of the continuity set remains a continuity point of the Kester function exactly when the original function has value zero there.

This is an interesting feature of the construction.

A point can therefore be:

continuous for f

but

discontinuous for K[f].

That reversal is one of the things I am currently investigating.

Fixed points

Another natural question is whether there are functions that remain unchanged under the Kester transformation.

Suppose

K[f]=f.

At every point where f is continuous,

2f(x)=f(x).

Therefore,

f(x)=0.

At discontinuity points, the Kester function leaves the value unchanged automatically.

Thus:

\boxed{
K[f]=f
\quad\Longleftrightarrow\quad
f(x)=0\text{ at every continuity point of }f.
}

So the fixed points of the Kester transformation have a rather unusual property: every continuity point must have value zero.

Repeated Kester transformations

We can also apply the transformation repeatedly:

f,\quad K[f],\quad K^2[f],\quad K^3[f],\ldots

For a function that is continuous everywhere,

K[f]=2f,

so

K^2[f]=4f,

K^3[f]=8f,

and generally

K^n[f]=2^n f.

But for functions with discontinuities, the behaviour can be considerably more complicated because the continuity set can change after each transformation.

This leads to a natural question:

\boxed{\text{What happens to the continuity set under repeated Kester transformations?}}

That is something I would like to investigate further.

Why I find this interesting

The definition itself is very simple.

The interesting part is the feedback between the function and its own regularity.

Normally we take a function f and apply some rule to its values:

f(x)\longrightarrow F(f(x)).

The Kester function instead asks a question about f itself:

\text{Is }f\text{ continuous at }x?

The answer then determines what happens to f(x).

So the construction creates a relationship between:

\boxed{\text{value of the function}}

and

\boxed{\text{local regularity of the function}}.

I think this makes the Kester function an interesting object to explore, even though the definition is elementary.

Questions for further investigation

Some of the questions I am currently interested in are:

1. Can we completely characterize the continuity set of K[f]?
2. Can a Kester transformation create arbitrarily many new discontinuities?
3. What happens under repeated application of K?
4. Which functions are fixed points?
5. What happens if the factor 2 is replaced by an arbitrary number a?
6. Can the construction be generalized from continuity to other properties, such as differentiability?
7. Are there existing mathematical constructions equivalent to the Kester function under another name?

Final note

I am presenting this as an independent mathematical investigation, rather than claiming that the construction is already established or that it is definitively new.

The next stage is to investigate the mathematics more deeply and determine whether the Kester function leads to results that are genuinely new.

For now, this is simply the beginning of the investigation.

Kester Pembroke

Independent Mathematical Researcher