18.099b Problem Set 5a Due: Thursday, April 1st (in class or before).

This assignment is to write a short paper on the oscillation of a function. It should be typeset in Tex. Arrange and express the material discussed below in whatever way you think best. Be sure to define all the terms I have italicised (even if I have not defined them here).

Given a function f(x) and a real number c, we say that f is bounded about c if there exists some ϵ > 0 such that

(i) f is defined on [c − ϵ, c + ϵ] except possibly at c itself, and (ii) the set {f(x) : x ∈ [c − ϵ, c + ϵ], x ≠ c} is bounded from above and below. For any such ϵ let A_{ϵ} := {f(x) : x ∈ [c − ϵ, c + ϵ], x ≠ c}. The oscillation of f at c is defined to be the number

osc_{f}(c) := lim_{ϵ→0}(sup A_{ϵ} − inf A_{ϵ})

Show that this limit exists and hence osc_{f}(c) is well defined.

Show that for functions f bounded about a real number c, osc_{f}(c) = 0 if and only if f is continuous or has a removable singularity at c. Give two examples of non-zero oscillation, one where c is a jump singularity and one where c is an essential singularity.

An intuitive way to describe what it means for a function to be continuous at a point c would be to say: “f(x) can be made arbitrarily close to f(c) by making x arbitrarily close to c”. Fixing τ ≥ 0, give an analogous intuitive description of what it means for f to have oscillation τ at a point c. The idea is that oscillation measures the “amount” of discontinuity.

Suppose f is monotonically increasing and defined on an interval [a, b]. Note that f cannot have any removable singularities in [a, b], and that that for any c ∈ (a, b), f is bounded about c. Moreover, osc_{f}(c) = lim_{x→c^{+}} f(x) − lim_{x→c^{−}} f(x). Conclude

that for any τ > 0, the number of points in (a, b) at which f has oscillation greater

than or equal to τ is at most \frac{f(b) − f(a)}{τ}. Use this to show that f has at most countably many discontinuities in (a, b). (Recall that a set S is countable if there is a bijection between ℕ and S).