Proving Continuity of a Function at an Isolated Point

[xsw001]

[a]1. Homework Statement [/a]

Prove that a point xo in Domain is either an isolated point or a limit point of D.

[a]2. Homework Equations [/a]

xo in D is an isolated point provided that there is an r>0 such that the only point of domain in the interval (xo-r, xo+r) is xo itself.

[a]3. The Attempt at a Solution [/a]

Is there any other possibility if D={xo}, then xo is is certinaly not a limit point since single point of a set has no limit point, then it must be isolated point?

Another similar problem

Homework Statement

Suppose xo is an isolated point of D.
Prove that every function f:D->R is continuous at xo.

Homework Equations

xo in D is an isolated point provided that there is an r>0 such that the only point of domain in the interval (xo-r, xo+r) is xo itself.

Isolated point is NOT a limit point.

The Attempt at a Solution

Show for all epsilon>0, there exists a delta>0 such that if |x-xo|<delta, then |f(x)-f(xo)|<epsilon?

so I can just let r=delta isn't it?
so for every epsilon>0, there exists a delta=r>0 such that |x-xo|<r=delta, then |f(x)-f(xo)|<epsilon?

 

[lippyka]

Since xo is an isolated point, r>0 such that the only point of domain in the interval (xo-r, xo+r) is xo itself.This means that for any x such that |x-xo|<r, x=xo and since f is function, it's constant at xo which implies that |f(x)-f(xo)|=0<epsilon. Therefore, f is continuous at xo.