TRUE OR FALSE: Let [a, b] ⊂ R. The function f is integrable on [a, b] provided that f is uniformly continuous on [a, b].

Question

TRUE OR FALSE: Let [a, b] ⊂ R. The function f is integrable on [a, b] provided that f is uniformly continuous on [a, b].

Answer 1

TRUE

Here's the proof

By uniform continuity For every epsilon > 0, there is δ > 0 so that when x, y ∈ [a, b] and |x − y| < δ

then |f(x) − f(y)| < epsilon.

Let I ⊂ [a, b] be an interval of length less than δ. Then l.u.b.x∈ | f(x) | − g.l.b. x∈ I f(x)| < epsilon.

Therefore, for any partition P of [a, b] all of whose intervals are shorter than δ,

UP (f) − LP (f) < epsilon*(b − a).

Since this is true for every epsilon > 0, it means that

Iu,[a,b] (f) = Il,[a,b] (f).

So f is integrable on [a, b].