Suppose $\displaystyle f $ and $\displaystyle g $ are real valued functions that are Riemann integrable on $\displaystyle [a,b] $. Suppose further that $\displaystyle f(x) \leq g(x) $ for all $\displaystyle x \in [a,b] $ and that a strict inequality holds for at least one point in $\displaystyle [a,b] $. Prove that $\displaystyle \smallint_{a}^{b} f < \smallint_{a}^{b} g $ need not hold. If we add a continuity condition to $\displaystyle f $ and $\displaystyle g $, does the above inequality hold?

**Proof**. We know that $\displaystyle \smallint_{a}^{b} f \leq \smallint_{a}^{b} g $. Define $\displaystyle f $ and $\displaystyle g $ as follows:

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

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

Here we have $\displaystyle c \in [a,b] $ and $\displaystyle K < L $. Then $\displaystyle \smallint_{a}^{b} f = \smallint_{a}^{b} g = 0 $. $\displaystyle \diamond $

If we had a continuity condition, then I think the above inequality will hold? Do we use the following fact: There exists an open interval $\displaystyle (c,d) $ in $\displaystyle [a,b] $ such that $\displaystyle x \in (c,d) $ and $\displaystyle f(u)>0 $ for all $\displaystyle u \in (c,d) $? Is this correct?