Integrability of a continuous function over an open interval

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Consider the following statement:

Let $I$ be a bounded open interval and let $f$ be a real function that is continuous on $I$. Then, $f$ is integrable on $I$

This statement is false, but I cannot understand why. I thought that a continuous function on a bounded interval will be bounded, which seems to imply integrability.

Edit: Also, why is it then that $\lvert\int f(x)dx\rvert$ from $a$ to $b$, where $a,b$ are in the interval $I$, is strictly bounded?

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