Proposition: Continuity preserves limits

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