If $\lim_{h\to 0}[f(x+h)-f(x+h)]=0$ for all $x$ then $f$ is not necessarily continuous

Question

Let $f : \mathbb{R}\to \mathbb{R}$ be a function. If, for all $x \in \mathbb{R}$, it holds that $$\lim_{h \to 0} [f(x+h) - f(x+h)] = 0$$

then this would imply that $f$ is continuous ?

Solution:

No.

But I don't know how to justify this result.

Answer 1

Answer with $x+h$ in second term changed to $x-h$. Take $f(x)=x$ for $x \neq 0$ and $f(0)=1$.

Answer 2

Well it's actually kind of silly. $f(x+h) - f(x+h) =0$, with no limit required.