how to know when the function is continuous

Expand sin(a+x) as a series in x:  

sin(a+x) = sin(a) + x*cos(a) + higher order terms involving multiples of x.

Similarly:

sin(a-x) = sin(x) - x*cos(a) + higher order terms

Therefore, sin(a+x) - sin(a-x) = 2x*cos(a) + higher order terms

(sin(a+x) - sin(a-x))/x = 2cos(a) + higher order terms.

The higher order terms all contain multiples of x, so when x goes to zero, these go to zero and we are left with:

$$\lim_{x \rightarrow 0}\frac{\sin(a+x)-\sin(a-x)}{x}=2\cos(a)$$

I do not know this expansion Alan.  

Thanks Alan.