Suppose we have a continuous function $f:[a,b]\to\mathbb{R}$. We know that $f$ is differentiable on $(a,b]$ and that there is a finite limit:
$\lim_{x\to a^{+}}f^{\prime}(x)$
Can we prove that $f$ has a right derivative in point $a$?
Suppose we have a continuous function $f:[a,b]\to\mathbb{R}$. We know that $f$ is differentiable on $(a,b]$ and that there is a finite limit:
$\lim_{x\to a^{+}}f^{\prime}(x)$
Can we prove that $f$ has a right derivative in point $a$?
For every $x>a$ there exists $\xi \in (a,x)$ such that $f(x)-f(a)=f'(\xi)(x-a).$ If $x \to a$, then $\xi \to a$, and ...
$f'_+(a)=\lim_{x \rightarrow a^+} \frac{f(x)-f(a)}{x-a}.$ By continuity of $f$, we have here indefinite symbol $\frac{[0]}{[0]}$. Using de L'Hospital rule we obtain
$f'_+(a)=\lim_{x \rightarrow a^+} f'(x).$ (The last limit exists by assumption).