## Proposition 6.2.3: Continuity preserves Limits

If

*f*is continuous at a point*c*in the domain*, and***D***{x*is a sequence of points in_{n}}*converging to***D***c*, then*f(x) = f(c)*.
If
* f(x) = f(c)*
for every sequence *{x _{n}}* of points in

*converging to*

**D***c*, then

*f*is continuous at the point

*c*.

### Proof:

The proof is very similar to the previous result about the equivalence of the two definitions of limits for a function. It is therefore left as an exercise. It would be good practice to see if you can modify the previous proof and adapt it to this result.