
continuity at a point
using the definition: $\displaystyle \forall \epsilon >0, \exists \delta >0, \forall h, \ h<\delta \ : f(a+h)f(a)<\epsilon$.
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 $\displaystyle h=\min(\delta/2,0.5)$ Is there a general method?
Thanks

Let $\displaystyle \epsilon<1$. Then for all $\displaystyle h<1$, $\displaystyle f(3+h)f(3)=03=3>\epsilon$. In this case, its probably easier to look at the onesided 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.