Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 31 / 1 / b / Solution

Take $m=\lceil (b-a)/\epsilon \rceil$. The ceiling, as printed in the PDF, makes the intervals cover the bounded range of f. If $a=b$, use the single set $\mathcal{X}$ instead, since f is constant. By continuity their preimages are closed. On every nonempty preimage $C_i$, the oscillation of f is at most epsilon. Hence for every $y\in C_i$,Taking the supremum over y in the second expression gives the bound for ${\sup }_{C_i}f-{\inf }_{C_i}I$; if that infimum is infinite, the bound is automatic. Apply the finite-cover estimate to obtainThis is the exponential-integral upper estimate in the bounded form of Varadhan lemma.