Prove Theorem 132 If fx is continuous on a b then f Ra bSolu

Prove Theorem 1-3-2: If f(x) is continuous on [a, b], then f R[a, b].

Solution

Here we want to show that function is Riemann integrable  

Given that function is continuous on [a,b]

by definition Riemann integrable we need to show that lower and upper integrable are same

so for every positive real no. epsilon we can find partition such that both lower and upper sum with respect to these partition are very close so by definition it follows that f is Riemann integrable

and alternatively by equivalent definition .function is Riemann integrable if it is contineous almost every where i,e   set of discontinuity has measure zero

But here allready f is contineous so set of continuity is empty set which has measure zero so we are done