# College Math Teaching

## February 24, 2014

### A real valued function that is differentiable at an isolated point

A friend of mine is covering the Cauchy-Riemann equations in his complex variables class and wondered if there is a real variable function that is differentiable at precisely one point.

The answer is “yes”, of course, but the example I could whip up on the spot is rather pathological.

Here is one example:

Let $f$ be defined as follows: $f(x) =\left\{ \begin{array}{c} 0, x = 0 \\ \frac{1}{q^2}, x = \frac{p}{q} \\ x^2, x \ne \frac{p}{q} \end{array}\right.$

That is, $f(x) = x^2$ if $x$ is irrational or zero, and $f(x)$ is $\frac{1}{q^2}$ if $x$ is rational and $x = \frac{p}{q}$ where $gcd(p,q) = 1$.

Now calculate $lim_{x \rightarrow 0+} \frac{f(x) - f(0)}{x-0} = lim_{x \rightarrow 0+} \frac{f(x)}{x}$

Let $\epsilon > 0$ be given and choose a positive integer $M$ so that $M > \frac{1}{\epsilon}$. Let $\delta < \frac{1}{M}$. Now if $0 < x < \delta$ and $x$ is irrational, then $\frac{f(x)}{x} = \frac{x^2}{x} = x < \frac{1}{M} < \epsilon$.

Now the fun starts: if $x$ is rational, then $x = \frac{p}{q} < \frac{1}{M}$ and $\frac{f(x)}{x} = \frac{\frac{1}{q^2}}{\frac{p}{q}} = \frac{1}{qp} < \frac{1}{M} < \epsilon$.

We looked at the right hand limit; the left hand limit works in the same manner.

Hence the derivative of $f$ exists at $x = 0$ and is equal to zero. But zero is the only place where this function is even continuous because for any open interval $I$, $inf \{|f(x)| x \in I \} = 0$.