Construct counterexamples regarding bounded variation function

Question

1. Construct a function $f(x)\in C(0,1)$ such that $f'(x)$ exists almost everywhere and $f'(x)$ is Lebesgue integrable while $f\notin $ BV([$0,1$]).
2. Construct a function $f(x)\in C(0,1)$ such that $f'(x)=0$ almost everywhere for $[0,1]$ while $f\notin $ BV([$0,1$]). Furthermore, is there exists a function $f(x)\in C(0,1)$ such that $f'(x)=0$ almost everywhere for $[0,1]$ while for all $[a,b]\subset[0,1], f\notin$ BV([$a,b$]).

For 1., I want to try function of form $x^\alpha \sin(\frac{\pi}{x})$, the case $\alpha=1$ satisfies all the condition but $f'$ seems not Lebesgue integrable. I am tring other posibilities. For 2., I really have no ideas, the continuous condition is too strict for me.

Appreciate any help or hint! Please share your mind!