Is an increasing monotonic function in a closed interval also continuous?

[Lancelot1]

Hello all,

Is this statement true ? Is every increasing monotonic function in a closed interval also continuous ?

How do you prove such a thing ?

Thank you !

 

[HOI]

You don't prove it, it isn't true!

For example the function f(x)= x for \(\displaystyle 0\le x\le 1\), f(x)= x+1 for \(\displaystyle 1\le x\le 2\), and, in general, f(x)= x+ n for \(\displaystyle n\le x\le n+1\) for n an integer is monotone increasing for all x but is discontinuous at every integer.