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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?

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up vote 2 down vote accepted

If $L$ is continuous at $0$, using the hypothesis $f(x+h)-f(x)-L(h)$ converges to $0$ as $h\to 0$ and $\lim_{h\to 0}L(h)=0$ so $\lim_{h\to 0}f(x+h)-f(x)=0$ and $f$ is continuous at $x$.

Conversely, if $f$ is continuous at $x$, using the hypothesis again we get that $-L(h)=f(x+h)-f(x)-L(h)-(f(x+h)-f(x))$ and since $\lim_{h\to 0}f(x+h)-f(x)-L(h)=0$ and $\lim_{h\to 0}f(x+h)-f(x)=0$ we get that $\lim_{h\to 0}L(h)=0$. So we can find $\delta>0$ such that if $|h|\leq\delta$ then $|L(h)|\leq 1$, so $L\left(\frac{\delta }{2|h|}h\right)\leq 1$ and $L(h)\leq \frac{|h|}{2\delta}$, which proves continuity of $L$.

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