# Implied continuity

1. Nov 2, 2009

### filter54321

1. The problem statement, all variables and given/known data
Let f be a real valued function whose domain is a subset of R. Show that if, for every sequence xn in domain(f) \ {x0} that converges to x0, we have lim f(xn) = f(x0) then f is continuous at x0.

2. Relevant equations
Book definition of continuity:
"...f is CONTINUOUS at x0 in domain(f) if, for every sequence xn in domain(f) converging to x0, we have limnf(xn)=f(x0)..."

3. The attempt at a solution
The statement lim f(xn) = f(x0) would suggest that f(x0) exists, so leave that part of continuity aside for now.

What's the trick to get from domain(f) \ {x0} to domain(f) to satisfy the defintion?

2. Nov 2, 2009

### n!kofeyn

Use the limit hypothesis to help show the standard epsilon-delta definition of continuity is satisfied at x0. You will use the epsilon-delta definition of the limit.