In order to use part 1 of the Fundamental Theorem of Calculus in the movie
above, it is
required that the function f(g(x)) g'(x) be continuous
on the interval [a, b]. One way we can guarantee this is to
assume that
g' is continuous on [a, b] and
f is continuous on the range of g which includes the
interval with endpoints g(a) and g(b).