suppose that 'g' is continious at c and f is continious at g(c) then
fOg is continious at c( please use continuity defintoin for proving
this theorem and please send me within oneday as it is very urgent or
me thank u)

Hi,

Thank you for your quesiton.

We have from the definition of continuity of g at c:
1-) lim g(x) = g(c)
      x->c

and the definition of continuity of f at g(c)
2-) lim f(x) = f(g(c))
      x->g(c)

Therefore we have:
lim f o g(x) = lim f(g(x))
 x->c           x->c

             = f(g(c))   (using 1-) and 2-)

This proves continuity of f o g at c.


There is another proof on page 5 of this document:
http://www.math.gatech.edu/~carlen/1507/notes/continuity.pdf

I hope this helps. If you need any clarifications please do not hesitate to ask.

Thanks.
endo