Calculus – Is There a Function $f: \mathbb R \to \mathbb R$ That Has Only One Point Differentiable?

Question

While studying complex variables, I could learn that $f(z)=|z|^{2}$ has only one point which is $z=0$ that $f$ being differentiable and $f$ being not differentiable at any other points.

Then, I was wondering if there is a function $f: \mathbb R \to \mathbb R$ that has only one point differentiable and not on any other points.

In intuition, it seems there are no such point!
However, I have no idea how I can prove this…

Additional question is that would there be any function $f: \mathbb R \to \mathbb R$ that has only one point continuous and not on any other points.

I think this is pretty interesting things to think about! 🙂

Answer 1

Let $$p(x)= \begin{cases} 0,& x\in\mathbb Q\\\\1,& x\in \mathbb R-\mathbb Q \end{cases}$$ Now take $f(x)=x^2p(x)$.