Questions on differentiability of $F(x)=\int_{0}^x f(t)dt$, where $f$ is locally integrable

Question

I'm going through Folland's Real Analysis, and has come to an impasse trying to come up with a couple examples in the topic of differentiability.

On $(\mathbb R, \mathcal B_{\mathbb R}, m)$, for locally integrable $f$, define $F(x)=\int_{0}^x f(t)dt$, $A_rf(x)=\frac{1}{2r}\int_{x-r}^{x+r}f(t)dt$.

1. Give an example where $\lim_{r\to 0^+}A_rf(x)=f(0)$, but $F$ not differentiable at 0.
2. Give an example where $F$ differentiable at $0$, but $\lim_{r\to0^+}A_r\lvert f-F'(0)\rvert(0)\ne 0$.

For 1, I believe I need an $f$ that is locally integrable, but is discontinuous at $x=0, but I haven't found a working example yet.

For 2, here I think I need an $f$ locally integrable, continuous at $0$ but $x=0$ is not in the Lebesgue set of $f$. I think I have an understanding on what properties I'm looking for, but I'm not sure if these are right, or if how to come up with examples.

Thanks!

Answer 1

I believe the example one need $x=0$

1. if $x\ne0,$ $f(x)=sgnx$, $f(0)=0$
2. If F′(0) exists, then $\lim_{x\to 0}\int_0^xf(t)dt/x$ exists, hence $\lim_{x\to 0}\int_{-x}^xf(t)dt/2x$ exists and equals $\lim_{r\to0+}A_rf(0)$.