In proving the "Invariance Under Translation of Definite Integrals" by substitution, is the continuity of the integrated function mandatory?

Question

In the beginning, I shall apologise if my question has been answered before (I can't find such an answer), and if my question is naive (I am not of a Mathematical or even scientific background, I am just trying to cut my way through Apostol Calculus for the sake of getting my weak brain around some calculus.) Required is a proof of the "Invariance Under Translation of Definite Integrals", by substitution. This property is stated as follows: THEOREM 1.18: If f is integrable on [a,b], then for every real c we have $$\int_a^bf(x)=\int_{a+c}^{b+c}f(x-c)dx.$$ But then, to prove this by substitution, I would deploy the "Substitution Theorem for Integrals", which is stated as follows: THEOREM 5.4: Assume g has continuous derivative g$\prime$ on an open interval I. Let J be the set of values taken by g on I and assume that f is continuous on J. Then for each x and c in I, we have $$\int_c^xf[g(t)]g\prime(t)dt=\int_{g(c)}^{g(x)}f(u)du.$$ Such deployment would require the continuity of f. The solution to Thomas Calculus to a like question did the same without assuming the continuity of f. Nor other resources. They applied it, assuming only that f is integrable. So, is continuity mandatory to apply THEOREM 5.4, or am I missing a point?

Answer 1

There are generalization of Theorem 5.4 which don't require continuity. If $f$ is any integrable function, that relationship holds (the assumptions on $g$ are important though). I don't think it is easy to prove though without more advanced techniques outside of the case that $f$ is continuous.