One of my instructors provided me with the following Change of Variable Theorem:
Let $a < b$.
Let $f$ be a continuous function.
Let $g$ be a function with a continuous derivative on $[a, b]$.
Assume the range of $g$ on $[a, b]$ is contained in the domain of $f$.
Then,
$$\int_a^b f(g(x)) g'(x) \, dx = \int_{g(a)}^{g(b)} f(u) \, du$$
I was wondering if it's sufficient to say "let $f$ be a continuous function" as above.
Shouldn't we specify $f$ to be continuous on $[a, b]$? Because a function being continuous means it's continuous on its domain, and if the domain is not specified, there's no guarantee that it'll be continuous on $[a, b]$, and therefore the last line would not be a well-defined expression.
When we write “$f$ is continuous” it is implied that we mean $f$ is continuous on its domain.
Note that it is not sufficient to say $f$ is continous on $[a,b]$. Rather, we need $f$ to be continuous on $g([a,b]) =\{y\mid y=g(x), x\in[a,b]\}$.