Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be continuous and let $a$ be a nonzero real number. Show that the function
$F(x)=\frac{1}{2a}\int_{-a}^{a}{f(x+t)dt}$ is differentiable and has continuous derivative.
My thoughts were that
$\frac{d}{dx}(\frac{1}{2a}\int_{-a}^{a}{f(x+t)dt})= \frac{1}{2a}\frac{d}{dx}(\int_{-a}^{a}{f(x+t)dt}) = \frac{1}{2a}(f(x+a)-f(x-a))$, which is continuous.
Is this correct? It seemed a little too easy.