Show that $\chi_{[0,t]}$ is not differentiable in $L^1$ norm.

Question

Here is my solution:

$$\lim_{h \to 0}\int{|\frac{\chi_{[t,t+h]}}{h}-g|}=0$$

$|\frac{\chi_{[t,t+h]}}{h}-g|\to |g| $ pointwise and so by Fatous $\int|g|\leq0$ and so $g=0$ a.e. This is a contradiction as $\lim_{h \to 0}\int{|\frac{\chi_{[t,t+h]}}{h}|}=1$. Is this correct?