Well, if he is asking this, he probably doesn't know about "Lipschitz"- or "uniformly continuous"!
CrazyCat87, The definition of f(x) continuous at x= a is that f(a) exist, exist, and . Typically, showing that the two are equal requires showing they exist so that is enough.
Here, if x> 0, f(x)= |x|= x. If a> 0, we can take so that as long as , x> 0 and will be true as long as is less than the smaller of a/2 and .
If x< 0, f(x)= |x|= -x. If a< 0, we can take so that as long as , x< 0 and will be true as long as is less than the smaller of -a/2 and .
Finally, if a= 0, |f(x)- f(a)|= | |x|- 0|= |x|. We can make that less than by taking .