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

The continuity of f makes its prescribed interval preimage B open, and $\hat{x}\in B$. On B we have $f(x)>f(\hat{x})-\epsilon$, soThe open-set lower bound of the large deviation principle therefore yieldsThus the claimed lower bound holds at every chosen point. When $I(\hat{x})=\infty$ its right side is minus infinity and the assertion is immediate.