Let $f$ be an integrable function on $[a, b]$.
Show that $f$ remains integrable on $[a,b]$ if we alter it at one single point and that this alteration does not change the value of the integral $\int^b_a f(x) dx$.
In lecture we have proven this statement by splitting up the altered $f^*$ into the old $f$ function and an indicator function. This approach seems perfectly reasonable for me.
However, I am curious about whether it is possible to show the statement without an indicator function.
Any comments or thoughts on this are welcome.