Example 4.22 : Let f : ℝ → ℝ be given by f(x) = x for x ≠ 0, f(0) = 1.

Then lim f(x)=0 [x→0]. For given ε > 0, put δ = ε. If 0 < |x − 0| < δ, then
|f(x) − 0| = |x| < ε, as required.To emphasize further that f(a) is irrelevant to
the existence or value of lim f(x) [x→a], it note that Definition 4.21 makes sense
even if f(a) is not defined—it is enough to assume that f is defined on some
subset A ⊆ ℝ, where A contains numbers arbitrarily close to but not equal to a.