Bonuded Variation and derivatives

Question

Wanted to ask if anyone could clarify the relationship between bounded variation and the derivative of the function?

Is there a particular theorem that states this relationship?

Thank you!

Answer 1

If the left-hand side is $V_a^b(f, P) = \sum_{k=0}^{n-1}|f(x_{k+1}) - f(x_k)|$ then, by comparing term by term, it suffices to show $$|f(b)-f(a)| \le \int_a^b |f'(t)| \, dt.$$ If you write down the fundamental theorem of calculus, wrap both sides with an absolute value, and push the absolute value under the integral, you'll arive at the above inequality.