Let $\displaystyle f: [a,b] \to \mathbb{R} $ be Riemann integrable on $\displaystyle [a,b] $. Suppose that $\displaystyle f(x) \geq 0 $ for all $\displaystyle x \in [a,b] $ and that $\displaystyle f $ is continuous on $\displaystyle [a,b] $. If $\displaystyle f>0 $ somewhere in $\displaystyle [a,b] $, prove that $\displaystyle \smallint_{a}^{b} f > 0 $.

Proof. If we look at the most extreme case in which $\displaystyle f $ is $\displaystyle 0 $ everywhere except at $\displaystyle c \in [a,b] $ then $\displaystyle \mathcal{R}(f,P) \geq 0 $. E.g. for the "extreme

case" we define $\displaystyle f $ as

$\displaystyle f(x) = \begin{cases} 0 \ \ \ \ \text{if} \ x \neq c \\ L \ \ \ \ \text{if} \ x = c \end{cases} $

where $\displaystyle c \in [a,b] $ and $\displaystyle L $ is a constant. So $\displaystyle \smallint_{a}^{b} f \neq 0 $. Thus $\displaystyle \smallint_{a}^{b} > 0 $. $\displaystyle \diamond $

Is this correct?

If $\displaystyle f $ were not continuous then the above statement would be false, right?