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Unassuming

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If f is continuous at some p in R, prove that f is continuous at every point of R.

Proof: Suppose f(x) is continuous at p in R. Let p in R and e>0. Since f(x) is continuous at p we can say that for all e>0, there exists a d>0 such that if 0<|x-p|<d implies that |f(x)-f(p)|< e/2.

I know that |f(x)-f(y)| = |f(x)-f(p)+f(p)-f(y)| <= |f(x)-f(p)|+|f(y)-f(p)|.

I also know that |x-y|=|(x-y+p)-p| <= |x-p|+|y-p|.

Help: I am confused on how to show that f is cont. at every point in R. I appreciate any help little or big! Thank you