In a homework sheet with true or false questions I have found the folowing statement
$\displaystyle \lim_{h \to 0}\left[ f(x+h)-f(x) \right]=0$
Is this True OR False?
At the first sight of it True seems the right answer but then this came to my mind.
If $f$ is not continuous at $x$ then
$$\lim_{x \to x_0}f(x)\neq f(x_0)$$
so applying the same logic to our example, suggests that False is the right answer
It seems a silly question but I need a second opinion.
Best Answer
In general, this is false. Consider $f(x)=\frac{1}{x}$ if $x\neq 0$ and $f(x)=0$ otherwise.
Then $$\lim_{h\to 0}[f(h)-f(0)]=\lim_{x\to 0}\frac{1}{x}$$