Plane Geometry - Bruce E. Shapiro

Section 20The Continuity AxiomRecall that a function f is continuous if ∀ɛ > 0, ∃δ > 0 such that|x − y| < δ ⇒ |f(x) − f(y)| < ɛTheorem 20.1 Let [a, b] and [c, d] be closed intervals of real numbers andlet f : [a, b] ↦→ [c, d] be a function. If f is strictly increasing and onto, thenf is continuous.Proof. Let ɛ > 0 be given (and sufficiently small) and let x ∈ (a, b).Since f is strictly increasing, c < f(x) < d for all x ∈ (a, b). We takec < f(x) − ɛf(x) + ɛ < dIf we did not start with an ɛ that was sufficiently small than reduce thevalue of ɛ to, say, (1/2) min[|f(x) − c|, |f(x) − d|]Since f is onto, there exist numbers x 1 , x 2 ∈ [a, b] such thatf(x 1 ) = f(x) − ɛ and f(x 2 ) = f(x) + ɛChoose δ = 1 2 min [|x − x 1|, |x − x 2 |]Then x 1 < x − δ and x 2 > x + δSuppose that |x − y| < δ. Thenx − δ < y < x + δ99