Originally Posted by

**southprkfan1** Well, I'll tell you off the bat I have know idea what "strike" represents

But, here's what I was working on before, and you can feel free to give me your thoughts.

Suppose f is not continuous:

Then, there exists e>0 and a xn (n=1,2,3,...) in M where

dM(xn, a) < 1/n but dN(f(xn), f(a)) > e

clearly (xn) converges to a, so by assumption (f(xn)) is a convergent sequence. Say it converges to f(x)

Thus, for n large enough, we have dN(f(xn), f(x))< e + d(f(x),f(a)) and dM(an, a)<1/n

but d<(xn, a) < 1/n --> e < dN(f(xn), f(a)) <= d(f(xn), f(x)) + d(f(x),f(a))

--> dN(f(xn), f(a)) > e - dN(f(x),f(a))

--> e + d(f(x),f(a)) > e + dN(f(x),f(a))

--> d(f(x),f(a)) > 0

--> f(x) /= f(a)

And I'm stuck

(well, not really, I could use the last posters hint to say f(x) = f(a), but in that case all this crap above would be unnecessary)