Is there a measure for how discontinuous is a given function?

Question

Suppose $X\subseteq\mathbb{R}$ and $Y\subseteq\mathbb{R}$ are arbitrary sets.

Motivation:

I want to measure how discontinuous is a given function $f:X\to Y$. The measure should range from zero to positive infinity: e.g., when $f$ is continuous the measure should be zero; however, when $f$ is "hyper-discontinuous" the measure should be positive infinity.

When the measure of how discontinuous is a function has a value between zero and positive infinity, the more "disconnected" the graph of $f$, the higher the value of the measure.

Question:

Does a measure of how discontinuous $f$ is exist? If not, how do we create one?

Answer 1

This way to measure the total discontinuity of a function is based on the following definition from Spivak's "Calculus on Manifolds" (p. 12-13): Let $f:\mathbb{R}\to\mathbb{R}$ be bounded. Then define the oscillation of $f$ at $a\in\mathbb{R}$, $o(f,a)$, by $$o(f,a)=\lim_{\delta\to 0} M(f,a,\delta)-m(f,a,\delta),$$ where $M(f,a,\delta)$ and $m(f,a,\delta)$ are the max and min of $f$ on the open ball centered at $a$ of radius $\delta$, respectively (if we allow $f$ to be only bounded from below or from above, then we can extend the definition by allowing $o(f,a)$ to possibly be infinite. If $f$ is unbounded, it could be that the oscillation is not well defined, for example looking at $o(f,0)$ for a function $f=1/x$ on $\mathbb{R}\setminus \{0\}$).

This limit exists since $o(f,a)$ decreases as $\delta$ decreases, and the quantity in the limit is bounded below by zero. As Spivak notes, $o(f,a)$ is a way to measure how discontinuous a function is at a point. The above definition can easily be lifted to any bounded function $f:X\to Y$ for $X,Y\subset \mathbb{R}$ just by restricting everything to the subspace topology on $X$. Then one can show that $f$ is continuous at $x\in X$ if and only if $o(f,x)=0$ (with this in mind, one could also extend this definition to measure the level of continuity on an arbitrary metric space instead of just $\mathbb{R}$).

Now we make some additional observations about the oscillation $o(f,x)$. First, observe that $o(f,x)$ is upper-semicontinuous, and therefore Lebesgue measurable (see here for more references and additional properties of the oscillation of a function). Moreover, a standard result in a measure theory class is that $D_f$, the set of discontinuities of a function $f$, is a Lebesgue measurable set.

Therefore, if one wants to obtain a measure of how discontinuous a function is, the following seems like a natural candidate in light of the above exposition: $$MC(f):=\int_{D_f} o(f,x)\:d\lambda(x),$$ where the integral is with respect to the Lebesgue measure (we restrict the domain of integration to be $D_f$ so that the definition makes sense even when the more natural domain $X$ is non measurable. If $X$ were measurable the two choices would coincide). Per the previous paragraphs, $MC(f)$ is well defined for any bounded function $f:X\to Y$.

I believe this definition also has all of the properties that were asked for by Arbuja. Since $o(f,x)=0$ iff $f$ is continuous at $x$, and because $o(f,x)\geq 0$, $MC(f)=0$ if and only if $f$ is continuous almost everywhere. Moreover, a function like $1_\mathbb{Q}$ has $MC(1_\mathbb{Q})=\infty$, since $o(1_\mathbb{Q},x)=1$ for all $x\in\mathbb{R}$ and $1_\mathbb{Q}$ is nowhere continuous.

We can also run a quick computation to see that this definition is able to qualitatively distinguish between functions where one is arguably "more continuous" than the other. Let $c>0$ and consider the function $$f_c(x)=e^{-cx^2}1_\mathbb{Q}(x).$$ Then even though $f_c$ is nowhere continuous, we could think of it as being more continuous than $1_\mathbb{Q}$ because $f_c$ rapidly approaches zero as $|x|\to\infty$, and so as $|x|$ becomes large, the function gets "closer" to being continuous in a loose sense. One can compute that $$o(f_c,x)=e^{-cx^2},$$ and from there it's straightforward to see that $MC(f_c)$ is a positive number which decreases to zero as $c\to\infty$ and increases to infinity as $c\to 0$. Note as $c\to 0$, $f_c$ approaches $1_\mathbb{Q}$ pointwise, while as $c\to\infty$, $f_c$ approaches 0 pointwise, which is continuous.

One limitation of this definition is that it cannot distinguish between $1_\mathbb{Q}$ and, say, $1_\mathbb{Q}/100$ in the sense these both have an infinite $MC$ if the domain $X$ on which we consider them has a subset of infinite measure. However even qualitatively I would posit this is not too much of an issue, since these functions are both discontinuous in a qualitatively identical way (both are discontinuous everywhere with constant oscillation). If you still wanted to find some way to compare the level of discontinuity between functions $f$ and $g$, something like $$\lim_{x\to\infty}\frac{MC(f|_{X\cap[-x,x]})}{MC(g|_{X\cap[-x,x]})}$$ could be useful (e.g. this would say that in some sense $1_\mathbb{Q}$ is 100 times more discontinuous than $1_\mathbb{Q}$), but I have not thought too much about it.

Answer 2

One possibility is thinking in terms of Sobolev-like spaces... which give a finer gradation on generalized functions (=distributions) than merely saying "not a classical function". Also, the Sobolev idea is compatible with differentiation, and so on.

Answer 3

Someone check if the the answer is wrong or over complex. If so, I hope there's a better version.

Motivation:

Let $X\subseteq \mathbb{R}$ and $Y\subseteq\mathbb{R}$ be arbitrary sets, where we define a function $f:X\to Y$.

Moreover, suppose $a,b\in\mathbb{R}$ and $\ell\subset\mathbb{R}^2$ is an arbitrary vertical line, where $\left\{(x,y)\in\mathbb{R}^2: a< x < b, y=0\right\}\cap \ell \not=\emptyset$.

When $f$ is continuous, $\ell$ interesects once with the graph of $f$, whereas if $f$ is the Dirichlet function, $\ell$ interesects with all limit points of the graph of $\left.f\right|_{[a,b]}$ twice. Therefore, we can subtract these values by one to get a measure of the discontinuity of $f$ on its domain.

However, when $f$ is "hyper-discontinuous", $\ell$ intersects with all limit points of the graph of $\left.f\right|_{[a,b]}$ zero times (when $f$ is discrete) or once ( when $\small{X=\mathbb{Q}\cap[0,1]}$) rather than infinitely. To fix this, $\ell$ should intersect with the intersection of the graph of $\left.f\right|_{[a,b]}$ and all limit points of the graph of $\left. f \right|_{[a,b]}$. The result would be zero, where the current measure is $0-1=-1$. Hence, since negative one is less than zero, we manipulate the current measure into the infimum of the emptyset or positive infinity.

Definition:

Suppose,

• $a,b\in\mathbb{R}$
• $G(a,b)$ is the graph of $\left.f\right|_{[a,b]}$
• $\mathbb{G}(a,b)$ is all limit points of $G(a,b)$
• $\ell\subset \mathbb{R}^2$ is a arbitrary non-vertical line, where $\left\{(x,y)\in\mathbb{R}^2: a< x < b, y=0\right\}\cap \ell \not=\emptyset$
• $|\cdot|$ is the absolute value
• $\#|\cdot|$ is the cardinality
• $\mathcal{D}$ is the measure of discontinuity, where $$\scriptsize{\mathcal{D}=\inf\left(\left\{\sup\limits_{a,b\in\mathbb{R}}\bigg(\inf\limits_{\text{x-intercept}(\ell)\in(a,b)}\bigg(\limsup\limits_{|\text{slope}(\ell)|\to +\infty}\bigg\{z-1: z=\#\Big|G(a,b)\cap\mathbb{G}(a,b)\cap\ell\Big|\bigg\}\bigg)\bigg)\right\}\setminus\{-1\}\right)}$$

Examples:

• When $f$ is continuous, $f$ has a discontinuous measure of zero
• When $f$ is the Dirichlet function, $f$ has a discontinuous measure of one
• When $f$ is the "hyper-discontinuous", $f$ has a discontinuous measure of $+\infty$
• When the graph of $f$ is dense in $\mathbb{R}^2$, $f$ has a discontinuous measure of $+\infty$

Answer 4

This might sound ridiculously naïve, but what about simply counting the amount of discontinuity points?

Imagine $f_1 : \mathbb{R} \to \mathbb{R} : f_1(x) = 0$
The amount discontinuity points of $f_1(x)$ equals zero.

Imagine $f_2 : \mathbb{R} \to \mathbb{R} : f_2(x) = \text{sign}(x)$
The amount discontinuity points of $f_2(x)$ equals one ($x=0$).

Imagine $f_3 : \mathbb{R} \to \mathbb{R} : f_3(x) = \frac{1}{x^2-1}$
The amount discontinuity points of $f_3(x)$ equals two ($x=\pm 1$).

Imagine $f_4 : \mathbb{R} \to \mathbb{R} : f_4(x) = \frac{1}{\sin(\pi \cdot x)}$
The amount discontinuity points of $f_4(x)$ equals the amount of integer numbers ($\aleph_0$), as for every integer number, there's one discontinuity point.

Imagine $f_5 : \mathbb{R} \to \mathbb{R} : f_5(x) = 0$ if $x \in \mathbb {Q}$ and $f_5(x) = 1$ if $x \in \mathbb {R} / \mathbb {Q}$
The amount discontinuity points of $f_5(x)$ equals the amount of real numbers ($\aleph_1$), as for every real number, there's one discontinuity point.

Edit: what about "infinity_order()"?
You seem to say that:

• in case the number of discontinuity points is finite, you want zero.
• in case the number of discontinuity points is infinity, you want a higher number.

What if you create a function "infinity_order()", as follows: $\text{infinity_order}(n) = 0, \forall n \in \mathbb {Z}$
$\text{infinity_order}(\aleph_n) = n, \forall n \in \mathbb{N}$

Answer 5

For a function $f$ define $S_f$ to be the set of points for which $f$ is discontinuous. $S_f$ is measurable for any measurable function $f$. We can use the set density function $$d(S_f)(x)=\lim_{h \rightarrow 0} \frac{\mu(S_f \cap [x-h,x+h])}{2h}$$ Then the measure of discontinuity defined by $$\int_{-\infty}^{\infty} d(S_f) dx$$ (might?) meet what you're looking for.