Analysis | Part IA, 2003

Let f:[a,b]Rf:[a, b] \rightarrow \mathbb{R} be continuous. Define the integral abf(x)dx\int_{a}^{b} f(x) d x. (You are not asked to prove existence.)

Suppose that m,Mm, M are real numbers such that mf(x)Mm \leqslant f(x) \leqslant M for all x[a,b]x \in[a, b]. Stating clearly any properties of the integral that you require, show that

m(ba)abf(x)dxM(ba).m(b-a) \leqslant \int_{a}^{b} f(x) d x \leqslant M(b-a) .

The function g:[a,b]Rg:[a, b] \rightarrow \mathbb{R} is continuous and non-negative. Show that

mabg(x)dxabf(x)g(x)dxMabg(x)dxm \int_{a}^{b} g(x) d x \leqslant \int_{a}^{b} f(x) g(x) d x \leqslant M \int_{a}^{b} g(x) d x

Now let ff be continuous on [0,1][0,1]. By suitable choice of gg show that

limn01/nnf(x)enxdx=f(0),\lim _{n \rightarrow \infty} \int_{0}^{1 / \sqrt{n}} n f(x) e^{-n x} d x=f(0),

and by making an appropriate change of variable, or otherwise, show that

limn01nf(x)enxdx=f(0).\lim _{n \rightarrow \infty} \int_{0}^{1} n f(x) e^{-n x} d x=f(0) .

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