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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.

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# If F'(x) is bounded so is F(x)

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