Stuck on Analysis question dealing with Continuity of Set

[kbrono]

Homework Statement

Define f: [0,\infty) \rightarrow R by

f(x) = {0 if x is [0,1] and 1 if x is (1,\infty )

Homework Equations

I think if I can show that f is continuous on [0,1] and not continuous on every point of [0,1] then that will suffice. However I have now clue how to go about this, the definition of set continuity confuses me.

The Attempt at a Solution

I thought is we make S=[0,1] we can set an epsilon or chose a delta that will hold for some points but not for all but again I keep losing myself.

 

[Minimonster]

There is a jump discontinuity with this piecewise function. Select an epsilon greater than 1, there will be no delta around x=1 such that |x-1|< delta that implies |f(x)-f(1)|< epsilon.

EDIT: Sorry, that should be select an epsilon less than 1...