Total differentiability gives a local $O(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ increment bound and therefore continuity

If the source dimension is zero, the only admissible increment is $0$, so the stated bound holds with $C=\delta =1$ and continuity follows directly from the norm-induced metric. If $n=0$, every increment of $f$ is zero and the same bound and continuity hold. For the remainder assume both dimensions are positive.

By [L1], choose $\delta >0$ such that the remainder $r(h)=f(a+h)-f(a)-Df(a)h$ satisfies $\mathrm{\parallel }r(h){\mathrm{\parallel }}_2\le \mathrm{\parallel }h{\mathrm{\parallel }}_2$ whenever $0<\mathrm{\parallel }h{\mathrm{\parallel }}_2<\delta$.

If $K$ bounds $Df(a)$ as in [L2], the triangle inequality gives $\mathrm{\parallel }f(a+h)-f(a){\mathrm{\parallel }}_2\le (K+1)\mathrm{\parallel }h{\mathrm{\parallel }}_2$ for those $h$, and it also holds at $h=0$.

Given $\epsilon >0$, take $\mathrm{\parallel }h{\mathrm{\parallel }}_2<\min \{\delta ,\epsilon \mathrm{/}(K+1)\}$; step 2.1 is the metric continuity condition at $a$.