Exercise 9, § 1

Let $\mathscr{B}(\mathbf{N})$ be the space of bounded sequences $\mathbf{x} = (\chi_n)_{n \in \mathbf{N}}$ of real numbers, endowed with the norm $\| \mathbf{x} \| = \sup |x_n|$: give an example of a continuous map $t \mapsto \mathbf{f}(t) = (f_n(t))_{n \in \mathbf{N}}$ of $\mathbf{R}$ into $\mathscr{B}(\mathbf{N})$ such that each of the functions $f_n$ is differentiable for $t = 0$, but $\mathbf{f}$ is not differentiable at this point.