Continuity of an increasing interval surjection
= Continuity of an increasing interval surjection
A strictly increasing function from a closed real interval onto the interval between its endpoint values is continuous. Given a desired output tolerance, choose nearby output values on either side and use their preimages to trap nearby inputs. At the endpoints only one side is needed. Equivalently, a discontinuity of a monotone function creates a gap in its range, which is incompatible with surjectivity onto an interval.