TITLE: Prove $\int_0^1 f(g(x)) \, dx \leq \int_0^1 f(x)\,dx + \int_0^1 g(x)\,dx$
QUESTION [6 upvotes]: $f$ and $g$ are continuous functions on $[0, 1]$. $f$ doesn't decrease on this segment. Values of both $f$ and $g$ are in range $[0, 1]$. Prove following inequality
$$\int_0^1 f(g(x)) \, dx \leq \int_0^1 f(x)\, dx + \int_0^1g(x) \,dx$$
I couldn't think of any great idea to show that this inequality is actually satisfied by all functions with mentioned properties.
Thank you for your hints and help in advance!

REPLY [8 votes]: Let $x_0$ be such that :
$$\forall x \in [0,1], \quad f(x) - x \leq f(x_0) - x_0.$$
Then : $$\int_0^1 [f(g(x)) - g(x) ] \,dx \leq \int_0^1 [f(x_0) - x_0] \,dx = f(x_0) - x_0 $$
And : $$\int_0^1 f(x) \,dx \geq \int_{x_0}^{1} f(x) \,dx \geq (1 - x_0) f(x_0) \geq f(x_0) - x_0$$