## $f\in L^1(0,1)$, $f’\in L^1_{\rm loc}(0,1)$ exists, $xf’\in L^1(0,1)$. Does then $(xf)(0+) = 0$?

I have $$f\in L^1(0,1)$$ such that $$xf$$ is absolutely continuous on $$[0,1]$$. The latter is equivalent to $$f’\in L^1_{\rm loc}(0,1)$$ and $$xf’\in L^1(0,1)$$.

Does it follow from these assumptions that $$(xf)(0) = 0$$?

## Convergence of a recursion in $L^1(0,1)$

Let $$\mu_0>0$$, $$a>0$$, $$b>0$$, and $$f(t)$$, $$g(t)>0$$, $$p(t)$$ be some continuously differentiable functions over $$\mathbb{R}$$.

I am looking for various tools to study the stability of the following recursive formula $$h_n(t)=\frac{-\mu_{n-1} f'(t)}{g(\mu_{n-1} f(t))}, \; \forall t\in [0,1],$$

$$\mu_n=\frac{1}{a}\int_0^1h_n(t)\int_{t}^1p\left(b – \int_{0}^{s}h_n(r)dr\right)dsdt.$$

I specifically want to know if $$h_n(t)$$ is convergent in $$L^1(0,1)$$.