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

#### Lancelot

##### New member
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 !

#### Country Boy

##### Well-known member
MHB Math Helper
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.