Let f be a real function defined on (a, b). Prove that the set of points at which f has a simple discontinuity is at most countable. Hint: Let E be the set on which f(x-)<f(x+). With each point x of E, associate a triple (p, q, r) of rational numbers such that \text { (a) } f(x-)<p<f(x+) \text { (b) } a<q<t<x \text { implies } f(t)<p \text {, } \text { (c) } x<t<r<b \text { implies } f(t)>p \text {. } The set of all such triples is countable. Show that each triple is associated with at-most one point of E. Deal similarly with the other possible types of simple dis-continuities.

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