Newton Leibniz, Riemann integration and the fundamental theorem. Let $ f:[0,1]\to\mathbb{R} $ be an integrable function.
let $ a,b $ be real numbers, such that  $ a\leq f\left(x\right)\leq b $.
let $ g:[a,b]\to\mathbb{R} $ be an integrable function.
and let $ h:[0,1]\to\mathbb{R} $ be the function defined by:
$ h\left(x\right)=\intop_{a}^{f\left(x\right)}g\left(t\right)dt $
I have 2 questions about the situation here.

*

*if $ h $ is differentiable, does that imply that $ g $ is continuous ?


*if $ h $ is continuous, does it mean that $ f $ is continuous ? (over [0,1] )
Originally I had 2 more claims to decide if it's true or false.
It was :
if $ f $ is continuous - then $ h $ is continuous. I proved that this is correct.
and I had to decide if $ h $ is integrable function (without any additional assumption) and I also proved that this is correct.
But I cannot decide about the 2 claims that I mentioned here.
Any help will be great. Thanks in advance.

*

*Let $f$ be a constant function. Then $h$ is constant (hence differentiable) no matter how $g$ is.


*Let $g=0$. Then $h=0$ (hence it is continuous) no matter how $f$ is.
