If the function f (x) is continuous on [a, b] and has inverse function, is f (x) monotone on [a, b] and proved? emergency

If the function f (x) is continuous on [a, b] and has inverse function, is f (x) monotone on [a, b] and proved? emergency

Monotone. Proof: (counter proof) suppose that f (x) is not monotone in [a, b], that is, the function must be increasing or decreasing, and the function is continuous, then at least exist that X1 belongs to [a, b], X2 belongs to [a, b], and X1 is not equal to X2, such that: C = f (x1) = f (x2); suppose that the inverse function of F (x) is g (x), then according to the definition, there are g (c) = X1 and G (c) = X2, that is, X1 = x2