Question

If f: [a,b] → E’ is a continuous function, then prove that f is Riemann integrable on
[a,b]. Discuss its converse by giving suitable examples.

Answer 1

Answer:

every continuous function is reimann integrable.Recall the definition of Riemann integral. To prove that

f is integrable we have to prove that

limδ→0+S∗(δ)−S∗(δ)=0

Since S∗(δ) is decreasing and S∗(δ) is increasing it is enough to show that given

ϵ>0 there exists δ>0

such that

S∗(δ)−S∗(δ)<ϵ.

So let ϵ>0 be fixed.

By Heine-Cantor Theorem

f is uniformly continuous i.e.

∃δ|y|<δ

⇒|f(x)−f(y)|<ϵb−a.

Let now

P

be any partition of

[a,b] in (δ)

i.e. a partition

{x0=a,x1,…,xN=b} such that xi+1−xi<δ

. In any small interval [xi,xi+1 ] the function f

(being continuous) has a maximum Mi and minimum mi.

Since f is uniformly continuous and xi+1−xi<δ

we have Mi−mi<ϵ/(b−a).

So the difference between upper and lower Riemann sums is∑i Mi(xi+1−xi)−∑i mi(xi+1−xi)≤ϵb−a∑i (xi+1−xi)=ϵ.

This being true for every partition P in C(δ)

we conclude that S∗(δ)−S∗(δ)<ϵ