Show Continuity of f at x0 via Implied Convergence

[filter54321]

Homework Statement

Let f be a real valued function whose domain is a subset of R. Show that if, for every sequence x_n in domain(f) \ {x₀} that converges to x₀, we have lim f(x_n) = f(x₀) then f is continuous at x₀.

Homework Equations

Book definition of continuity:
"...f is CONTINUOUS at x₀ in domain(f) if, for every sequence x_n in domain(f) converging to x₀, we have lim_nf(x_n)=f(x₀)..."

The Attempt at a Solution

The statement lim f(x_n) = 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 definition?

 

[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.