I was studying definite integrals and there is a property which is frequently encountered: $$\int_a^b f (x) \, dx=\int_a^b f (a+b-x) \, dx$$
Well, in one of my textbooks it is written that $f (x) $ should be continuous in $(a,b) $ for this property to hold good and in another textbook no such thing is mentioned. Also, in the textbook in which the condition for $f $ to be continuous is mentioned, the fact that $f $ is continuous is not used anywhere in the proof in the book.
So what is the real thing? Does $f $ need to be continuous or not?
According to me $f $ need not be continuous because the graph of $f (x) $ and the graph of $f (a+b-x) $ are just flipped about $x=\frac {a+b}{2}$
Best Answer
It is not necessary for $f$ to be continuous, but it is possible that an introductory textbook gave a definition of the integral that is applicable only when $f$ is continuous, in which case that may be the reason why the proposition is stated in that way.