Differentiability of $ -f$

Question

If a function, defined for all $x$, is differentiable on all $x$, $x$ being a real number. Would $-f$ also be differentiable on all $x$ , $x$ being a real number? I can’t think of a counterexample so I think this statement would be true. Can someone correct me if I’m wrong?

Answer 1

Sure. Consider $cf(x)$ for $c \in \mathbb{R}$. The limit

$$\lim_{h \to 0}\frac{cf(x+h) -cf(x)}{h}=c\lim_{h \to 0}\frac{f(x+h)-f(x)}{h}=cf'(x)$$

exists. Take $c=-1$.

Answer 2

$\ f$ is differentiable at all $\ x$. That means its graph does not have any sharp point and is continous for all $\ x$. So even if we flip the curve about $\ x$ axis which is$\ -f$, it will remain continous at every point and smooth. Thus it is differentiable too.