How to solve the following:
Let $f:R\rightarrow R$ be a continuous function, $h\in R-\{0\}$ and $F_h:R\rightarrow R$ defined with $F_h=\frac{f(x+h)-f(x)}{h}$. Prove that
a)$F_h\in D'(R)$,
b)$\lim_{h\rightarrow 0} F_h=f'$ in $D'(R)$.
Thanks in advance.
As Vobo pointed out, a) is obvious since $F_h$ is continuous (and hence $L^1_{loc}$) for every $h \neq 0$.
For b), let $\phi$ be a test function. Then
\begin{align} \langle F_h, \phi\rangle &= \int_{-\infty}^\infty \frac{f(x+h)-f(x)}{h} \phi(x)\,dx \\ &=\int_{-\infty}^\infty \frac{f(x+h)}{h} \phi(x)\,dx - \int_{-\infty}^\infty \frac{f(x)}{h} \phi(x)\,dx \\ &= \int_{-\infty}^\infty \frac{f(x)}{h} \phi(x-h)\,dt - \int_{-\infty}^\infty \frac{f(x)}{h} \phi(x)\,dx \\ &= -\int_{-\infty}^\infty f(x) \frac{\phi(x)-\phi(x-h)}{h}\,dx \\ &\to -\int_{-\infty}^\infty f(x) \phi'(x)\,dx = \langle f',\phi \rangle \end{align} as $h\to 0$, for example by dominated convergence.