1. The problem statement, all variables and given/known data Let f:[0,1] →ℝ be a continuous function that does not take on any of its values twice and with f(0) < f(1), show that f is strictly increasing on [0,1]. 2. Relevant equations 3. The attempt at a solution Assume that f is not strictly increasing on [0,1]. Therefore there exists a,b,c between 0 and 1. where 0 < a < b < c < 1. such that f(0) < f(a) < f(b) and f(c) < f(b) < f(1). By the intermediate value theorem there exist a x between (a,b) and a y between (b,c) for there exists a k where f(a) < k < f(b) and f(c) < k < f(b) and f(x) = k and f(y) = k. Contradiction Therefore f is strictly increasing on [0,1]. Does this proof work or am I missing something again?