Yet another proof function continuity related

[bluevires]

http://www.uAlberta.ca/~blu2/question1.gif


hey guys, I've tried this question and here's what I come up with, however I don't think this is anywhere near the right answer, but it does show the direction that I'm trying to work toward, I would appreciate any tips/help on how should I approach this question.

http://www.uAlberta.ca/~blu2/answer.gif

 

[AKG]

Let e be an arbitrary positive real number. Since f is continuous at L, there is a positive real d such that |L-x|< d implies |f(L)-f(x)| < e. Since the limit of g at a is L, then there exists a positive real c such that |x-a| < c implies |g(x)-L| < d. Is this enought of a hint?