# Thread: show these statements of riemann intergrable

1. ## 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)

2. 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 $\displaystyle f(x_0) > 0$ as some point $\displaystyle x_0 \in [a,b]$ then $\displaystyle f(x) > 0$ on $\displaystyle [a,b]\cap (x_0-\delta , x_0+\delta)$ for some $\displaystyle \delta > 0$. And then the integral cannot possibly equal to $\displaystyle 0$. 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.