Proving/Disproving a limit with absolute value

Question

Assume f: R -> R, if |f(x) − f(y)| = |x − y|, Then limx→∞ |f(x)| = ∞

I'm new to limits and was wondering how would I prove/disprove such a claim?

Answer 1

If $ |a| = |b| \Rightarrow a = \pm b$

so $ f(x) - f(y) = x-y \Rightarrow f(x) = x+c , c\in R$ (we get this by putting y = 0)

or

similarly $f(x) = -x +c$

and now we can easily verify that the limit is true...