1. ## Continuity

For function $f$ to be continuous at $a$, three conditions must be satisfied:

(1) $f$ is defined at $a$;
(2) $\lim_{x \to a} f(x)$ exists;
(3) $\lim_{x \to a} f(x)=f(a)$ exists;

Now say a function $g$ is defined as

$g(x) = \left \{\begin{array}{cc}0,&\mbox{if} x\not \in \mathbb{Z}\\1,&\mbox{if} x\in \mathbb{Z} \end{array}\right$.

Can we say $g$ is continuous $a$ where $a=2$ and $a \in \mathbb{Z}$?

Does $g$ satisfy the following?

(1) $f$ is defined at $a$ where $f(2)=1$;
(2) $\lim_{x \to a} f(x)$ exists;
(3) $\lim_{x \to a} f(x)=f(a)=0$ exists

?

2. Originally Posted by novice
Now say a function $g$ is defined as

$g(x) = \left \{\begin{array}{cc}0,&\mbox{if} x\not \in \mathbb{Z}\\1,&\mbox{if} x\in \mathbb{Z} \end{array}\right$.

Can we say $g$ is continuous at $a$ where $a=2$ and $a \in \mathbb{Z}$?

Does $g$ satisfies the following?
(1) $g(a)$ is defined.
Yes, $g(2)=1$.

(2) $\lim_{x \to a} g(x)$ exists;
Yes, $\lim_{x \to 2} g(x) = 0$.

(3) $\lim_{x \to a} g(x)=g(a)$
No. We have in (1) that $g(2)=1$ and in (2) $\lim_{x \to 2} g(x) = 0$. They're not equal, so condition (3) isn't satisfied. So g is not continuous at x = 2.

3. Now, change point $a$ to $a \not \in \mathbb{Z}$.

Then $g(x)$ is contiunous at $x= a, \forall a \not \in \mathbb{Z}$.

Yah?

4. Yes indeed.