If f is defined on [a,b] and for every x in [a,b] there is a d_x such that if is bounded on [x-d_x, x+d_x]. Prove that f is bounded on [a,b]. This question seems very odd. If every point, and indeed the neighbourhood of every point is bounded, then of course the function itself must be bounded. Of course I doubt this can be passed as a proof, so any suggestions would be helpful.