Thread: show these statements of riemann intergrable

Show these statements are true for continuous function f, but false for Riemann integrable functions f.

1. If f:[a,b] --> R is such that f(t)>=0 for all t in [a,b] and
integral( f(t), a, b) =0 then f(t)=0 for all t in [a,b]

2. integral(f(x), a, t) is differentiable and the derivative = f(t)

Originally Posted by szpengchao

1. If f:[a,b] --> R is such that f(t)>=0 for all t in [a,b] and
integral( f(t), a, b) =0 then f(t)=0 for all t in [a,b]

Use the continuity property, if as some point then on for some . And then the integral cannot possibly equal to . When it is not continous just take a single point jump as a conterexample.

2. integral(f(x), a, t) is differentiable and the derivative = f(t)

The continous case follows from the fundamental theorem of calculus. Again use the same conterexample involving a point jump. The derivative of the integral at the point just is not equal to the original function at that point.