Real Analysis: Solution Given that f is continuous on [a,b] Also g is continuous and integral a to b fgdj =0 When composition of fg integral =0 and as both functions are continuous, this is possible only when fg =0 for allx in a,b Or f(x) =0 for each x in [a,b].

1. Real Analysis:
Solution
Given that integral a to b f dj exists
As x belongs to [a,b] integral a to x definitely exists for f.
If A'(x) = f(x) then by fundamental theorem of Calculus
A(x) = integral a to x of f dj, where x is in [a,b]
For every point x in [a,b]
lt A(x) exists and equal to A(x) as f is continuous.
Hence A is also continuous.