continuity of a function?

Question

Definition of a continuous function:

a) f(c) is defined

b) lim f(x), x-> c exists

c) lim f(x)= f(c) x->c

If conditions b and c hold true, is a automatically true?

Answer 1

Seems like it, I'm having troubles thinking of a situation where it wouldn't be.
Doug - part c wouldnt be correct though anyways since its a hole in the function. the limit as x approaches a f(x) would not = f(a)....he's asking if theres a situation where b and c are correct but a is not.

Answer 2

What you gave is the definition of continuity at a point (c), not continuity of the function. But the answer is the same.

No. C can't be true if f(c) isn't defined. Consider
(x²-1)/(x-1)
It is eveywhere continuous -except- at x=1 (where it is undefined). In fact, it looks just like the straight line
y=x+1
-except- at x=1, where it has a little 'hole' in it (where it's undefined)

Doug