Is there a function $f: \mathbb R \to \mathbb R$ that has only one point differentiable?
Solution 1:
Let $$p(x)= \begin{cases} 0,& x\in\mathbb Q\\\\1,& x\in \mathbb R-\mathbb Q \end{cases}$$ Now take $f(x)=x^2p(x)$.
Let $$p(x)= \begin{cases} 0,& x\in\mathbb Q\\\\1,& x\in \mathbb R-\mathbb Q \end{cases}$$ Now take $f(x)=x^2p(x)$.