If a scalar field is continuous at a point, then is it also differentiable at that point?

Question

Let $S$ be a non-empty subset of $\mathbb{R}^n$, let $\mathbf{a}$ be an interior point of $S$, and let $f \colon S \to \mathbb{R}$ be a scalar field such that $f$ is continuous at $\mathbf{a}$. Then is $f$ also differentiable at $\mathbf{a}$?

I know that the answer to this question is in the negative if $n= 1$. For example the function $f \colon \mathbb{R} \to \mathbb{R}$ defined by $$ f(x) \colon= \lvert x \rvert \ \mbox{ for all } x \in \mathbb{R} $$ is continuous at $x=0$ but is not differentiable at that point.

What if $n > 1$?

I also know that if $f$ is differentiable at $\mathbf{a}$, then $f$ is also continuous at $\mathbf{a}$.

Answer 1

Note that, if $x\in\mathbb R$, then $|x|=\sqrt{x^2}$. You can apply the same idea in $\mathbb{R}^n$. That is, you can define$$f(x_1,\ldots,x_n)=\sqrt{{x_1}^2+\cdots+{x_n}^2}.$$The function $f$ is continuous but not differentiable at $(0,0,\ldots,0)$. Indeed, if it was, then, if we consider the function $g\colon\mathbb{R}\longrightarrow\mathbb{R}^n$ defined by $g(x)=(x,0,0,\ldots,0)$, then $f\circ g$ would be differentiable too. But $f\bigl(g(x)\bigr)=|x|$ and therefore $f\circ g$ is not differentiable at $0$.

Answer 2

The same could be applied to for example take $$f(x_1,x_2,\cdots,x_n)=|x_1|+\cdots+|x_n|$$