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According to Wikipedia, the expected value of a continuous random variable is
$$E[X] = \int_{-\infty}^{\infty} xf(x) \mathrm{d}x.$$
Suppose $f$ is a function such as $f:\mathbb{R}\rightarrow (a,b)$. Is the expected value then undefined, since $f$ does not exist for $x < a$ or $x > b$?
If you define the function the way you did, $f(x)$ does exist for $x < a$ and $x>b$. Perhaps you mean a function $g:(a,b) \to \mathbb{R}$, where the domain is a bounded interval. In that case, $g(x)$ is assumed to equal $0$ for all $x \notin (a,b)$. An example is the Beta distribution, which is defined by a probability density function on the interval $(0,1)$ and $0$ elsewhere. The Beta distribution has an expected value, so the fact that the pdf is not defined for all elements of $\mathbb{R}$ does not imply the function does not have an expected value.
