Remark

If one endows the vector space of mappings from $I$ into $E$ and its subspace of differentiable mappings (cf. Gen. Top., X, p. 277) with the topology of simple convergence (or the topology of uniform convergence), the linear mapping $f \mapsto Df$ is not continuous (in general) for example, the sequence of functions $f_n(x) = \sin n^2 x / n$ converges uniformly to 0 on $\mathbf{R}$, but the sequence of derivatives $f'_n(x) = n \cos n^2 x$ does not converge even simply to 0.