Suppose f:A-->R is monotone (ACR: reals) and suppose the range of f is an interval, show f is continuous on A. By drawing a picture, I can see the conclusion. Since f is monotone, the only type of discontinuity it may have is a jump discontinuity. But since the range of f is an interval, this cannot happen. I would like to have a more consistent proof (analytical proof), I mean, I know that since f is monotone, for each point a of A, f(a+) and f(a-) exist. Now f(a+)> or eq. to f(a) and f(a-)< or eq. to f(a). Now how would I show that f(a+)< or eq. to f(a) and f(a-)> or eq. to f(a) using the fact that range(f) is an interval? Hence I would be able to conclude that f(a-)= f(a+)=f(a), therefore, f continuous on A.