Let $[a, b]$ be a compact interval of $\mathbb{R}$ and $f$ a function continuous on $[a, b]$ and differentiable on $]a, b[$, with real values. Suppose that $f'(x)$ has a finite limit $\ell$ as $x \rightarrow a^{+}$. Show that $f$ is right-differentiable at $a$ and specify the value of $f'(a)$.