MathDB
Bounded variation

Source: 26th annual VJIMC (2016), Category II, Problem 4

April 10, 2016
functionIntegralbounded variationdifferentiable functionreal analysiscollege contests

Problem Statement

Let f:[0,∞)→Rf: [0,\infty) \to \mathbb{R} be a continuously differentiable function satisfying
f(x)=∫x−1xf(t)dtf(x) = \int_{x - 1}^xf(t)\mathrm{d}t
for all x≥1x \geq 1. Show that ff has bounded variation on [1,∞)[1,\infty), i.e.
∫1∞∣f′(x)∣dx<∞.\int_1^{\infty} |f'(x)|\mathrm{d}x < \infty.
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