Represent $f(x) - f(y)$ as an integral Description
I've come across the following transition in a textbook of Convex Optimisation. I couldn't figure out what's going on so that I'd appreciate if anyone hits me with any hint!
Problem
Suppose $x, y \in \mathbb{R}^n$ and $f$ be a $\beta$-smooth convex function on $\mathbb{R}^n$;
Then the transition of interest goes as
$$f(x) - f(y) = \int^1_0 \nabla f(y + t(x - y))^{T} (x - y) dt $$
Additional Comment
This transition appears in the proof of the convergence rate of Gradient Descent for a smooth convex objective function $f$.

This is just the one variable fundamental theorem of calculus.
Let $\phi(t) = f(y+t(x-y))$.
Then $\phi(1) =\phi(0) + \int_0^1 \phi'(t)dt$, or equivalently,
$f(x) = f(y) + \int_0^1 D f(y+ s(x-y)) (x-y) ds $,
and since $Df(x)h = \langle \nabla f(x) h \rangle$, you have the desired result.
(The result is true for any real valued convex function defined on an open set containing the segment $[y,x]$ since it is then locally Lipschitz and differentiable ae.)
