Let $f : E \to F$ be a mapping between two infinite-dimensional normed vector spaces $E$ and $F$, and assume that there exists a linear mapping $L$ so that
$ \lim_{h \to 0} \frac{f(x + h) - f(x) - L(h)}{|h|} = 0. $
According to the author of my book, it is "easily verified" that $L$ is continuous at 0 if and only $f$ is continuous at $x$.
A mapping is said to be continuous if $\lim_{x \to x_0} f(x) = f(x_0)$, and I also know that a linear mapping is continuous at 0 if and only if there is a $c > 0$ such that $|L(v)| \leq c|v|$ for all $v$, but unfortunately I cannot "easily verify" the statement above... any hints?