We consider a function $f$ belonging to $\mathcal{B}_1$. We set $\bar{f}(r) = \frac{1}{2\pi} \int_0^{2\pi} f(r\cos t, r\sin t)\,\mathrm{d}t$. Prove that $\bar{f}$ is of class $C^1$ on $\mathbb{R}$.
We consider a function $f$ belonging to $\mathcal{B}_1$. We set $\bar{f}(r) = \frac{1}{2\pi} \int_0^{2\pi} f(r\cos t, r\sin t)\,\mathrm{d}t$.
Prove that $\bar{f}$ is of class $C^1$ on $\mathbb{R}$.