Special function $x: C[0,1]\to C[0,1]$

Question

I am looking for a function $x: C[0,1]\to C[0,1]$ with $\lim_{t^+ \to 0}\frac{x(t)}{t}$ exists, $\left\|\frac{x(t)}{t}\right\|_{C[0,1]}=\infty$ and $\left\|\,x(t)\,\right\|_{C[0,1]}<\infty$. I tried a lot but without success...

Answer 1

I suppose that you want a continuous function $x:[0,1]\longrightarrow\Bbb R$ ($x\in C[0,1]$ in the usual notation of $C[0,1]$).

If the limit exist, $t\mapsto x(t)/t$ has a removable discontinuity and $||x(t)/t||<\infty$. In any case, if $x$ is continuous, $||x||<\infty$.