I'm reading a paper "Formal Guarantees on the Robustness of a Classifier against Adversarial Manipulation" https://arxiv.org/abs/1705.08475
There, I try to understand proof for Theorem 2.1. It starts with saying "By the main theorem of calculus, it holds that:"
$$ f_j(x+\delta) = f_j(x) + \int_0^1\left<\nabla f_j(x+t\delta),\delta\right> d t, \text{ for } j = 1,\dots,K. $$
Where, $f_j(x): \mathbb{R}^d \rightarrow \mathbb{R}$, $\delta \in \mathbb{R}^d$
A simple way to formulate the fundamental theorem of calculus would be:
$$ f_j(x+\delta) = f_j(x) + \left<\nabla f_j(x+\delta),\delta\right> , \text{ for } j = 1,\dots,K. $$
But I do not understand where the integral comes from, and why is it correct?