Is non continuous function also not Bounded ?

[Lancelot1]

Dear all,

I am trying to figure out if a non continuous function is also not bounded. I know that a continuous function in an interval, closed interval, is also bounded. Is a non continuous function in a closed interval not bounded ? I think not, it makes no sense. How do you prove it ?

Thank you !

 

[HOI]

You are making an error in logic. The opposite of "if A then B" is NOT "if not A then not B". It is "if not A then we don't know anything about B"!

For example the functions f(x)= x for \(\displaystyle 0\le x\le 1\), f(x)= x+ 1 for \(\displaystyle 1\le x\le 2\) and g(x)= 0 if x is rational, g(x)= 1 if x is irrational are discontinuous (g badly discontinuous) but are bounded functions.