This is a proof for, if a function is continuous and has an inverse, then its inverse is continuous. I understand the proof, except I don't quite understand why f has to be strictly monotonic on $I_1$ for the interval $J_1$ to exist. Why must f be strictly monotonic to produce $J_1$? I understand a function must be one-to-one, and therefore strictly monotonic, to have an inverse, but why does it matter for $J_1$'s existance? Thanks for any insight you may have.
Just draw a picture of any increasing function on an interval which is not strictly increasing. There is a jump discontinuity at some point and you will see that there is a gap in the image of the interval.