Hello everybody, A few years ago i tried to join a mathematics department and in the relevant exams i came up against the following problem. I apologise beforehand if the statement of the problem is a little bit ambiguous because i do not remember it exactly. However, I am sure you will get the point. 1. The problem statement, all variables and given/known data if the first derivative of the real function F(x) is continuous and bounded over the interval [a,b] (or (a,b) ?) , prove that F(x) also is bounded on the interval (a,b) (or [a,b] ?) and the vice versa. 2. Relevant equations So we can see that m =< F'(x) =< M. How can we get from this into the boundness of the F(x) without falling into pitfalls ? What about the vice versa ? Should we use the defintion or something else ? 3. The attempt at a solution I will not attempt to publish the solution I proposed because many of you may laugh.