Let $f,g:\mathbb{R}\to\mathbb{R}$ be two nondecreasing functions. Show that for any $a\in\mathbb{R},$ $b\in[f(a-0),f(a+0)]$ and $x\in\mathbb{R},$ the following inequality holds \[\int_a^xf(t) \ dt\geq b(x-a).\] Given that $[f(a-0),f(a+0)]\cap[g(a-0),g(a+0)]\neq\emptyset$ for any $a\in\mathbb{R},$ prove that for any real numbers $a<b$\[\int_a^b f(t) \ dt=\int_a^b g(t) \ dt.\] Note: $h(a-0)$ and $h(a+0)$ denote the limits to the left and to the right respectively of a function $h$ at point $a\in\mathbb{R}.$