# Continuity question

C.E
1. Let X be R be a finite set and define f : R $$\rightarrow$$ R by f(x) = 1 if x $$\in$$ X and f(x) = 0 otherwise. At which points c in R is f continuous? Give proofs.

3. I don't know how to start this, do you think it is ok to assume that X represents an interval of R? If not how can you possibly deduce the points continuity?

Homework Helper
Gold Member
X is finite. Meaning it contains a finite number of points. So X is certainly not an interval.

Try some concrete examples with increasing degree of complexity. Say X={0}. What then? (i.e., where is f continuous?) Now what if X={-1,1}, etc. If you've solved the problem in these two particular cases, then surely you can guess the answer to the general case and back your intuition with a proof.

C.E
Is this right?

The function is discontinous for all x in X and continuous elsewhere.
To prove discontinuity at x in X let x_1, x_2, ... x_n be the points in X then if we assume X_2 is the member of X closest to x_1. Then taking episilon =0.5 and only considering delta less than 0.5|x_1 - x_2| we prove discontinuity.

To prove continuity elsewhere we use a similar argument letting X_1 be the closest member of X to the point x not in X then setting delta= 0.5|X_1-x| completes the proof.

I think simpler is: since X is finite, there exist $\epsilon> 0$ such that the distance between any two points in X is greater than $\epsilon$.