# continuity at a point

• Apr 12th 2010, 04:22 AM
charikaar
continuity at a point
using the definition: $\forall \epsilon >0, \exists \delta >0, \forall h, \ |h|<\delta \ : |f(a+h)-f(a)|<\epsilon$.

f(x)=x if x is an integer.
f(x)=0 if x is not an integer. Prove that f is not continuous at=3 so we use the negation of the above definition.

how do we know $h=\min(\delta/2,0.5)$ Is there a general method?

Thanks
• Apr 12th 2010, 05:41 AM
Let $\epsilon<1$. Then for all $|h|<1$, $|f(3+h)-f(3)|=|0-3|=3>\epsilon$. In this case, its probably easier to look at the one-sided limits at an integer: since f is 0 as you approach an integer, the limit of f at any integer is 0, but f is not zero there, so it is discontinuous.