ma8_wk4_wednesday_notes_2010

ma8_wk4_wednesday_notes_2010 - MATH 8, SECTION 1, WEEK 4 -...

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MATH 8, SECTION 1, WEEK 4 - RECITATION NOTES TA: PADRAIC BARTLETT Abstract. These are the notes from Wednesday, Oct. 20th’s lecture, where we studied several different kinds of discontinuous functions. 1. Random Question Question 1.1. Can you find a function f : R R that’s continuous at every rational point q Q , but discontinuous at every irrational point a R \ Q ? 2. Discontinuity Proofs: A Lemma and a Blueprint How do we show a function is discontinuous? Specifically: in our last class, we described a “blueprint” for showing that a given function was continuous at a point. Can we do the same for the concept of discontinuity? As it turns out, we can! Specifically, we have the following remarkably useful lemma, proved in Dr. Ramakrishnan’s class on Wednesday: Lemma 2.1. For any function f : X Y , we know that lim x a f ( x ) 6 = L iff there is some sequence { a n } n =1 with the following properties: lim n →∞ a n = L , and lim n →∞ f ( a n ) 6 = L , and This lemma makes proving that a function f is discontinuous at some point a remarkably easy: all we have to do is find a sequence { a n } n =1 that converges to a on which the values f ( a n ) fail to converge to f ( a ). Basically, it allows us to work in the world of sequences instead of that of continuity; a change that makes a lot
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ma8_wk4_wednesday_notes_2010 - MATH 8, SECTION 1, WEEK 4 -...

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