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gracekerleyy
gracekerleyy
13.07.2019 • 
Mathematics

Let a, b e r. we learned that if f is continuous on (a, b) then f is integrable on [a, b]. we did not look at the proof of this, which is quite involved, in class. you will prove a weaker version of this theorem in this question. first we need a definition: definition: let c > 0. given f : [a, b] → r. we say f is c-pink iff "x € [a, b], vy € [a, b], \f (x) – f(y) 0. fix a c-pink function f : [a, b] → r. prove that f is integrable on (a,b). hint: let n e n. let pn be the partition dividing [a, b] into n equal sub-intervals. using the fact that f is c-pink, what simple expression can you conclude u(f, pn) - l(f, pn) is less than? conclude with the e-reformulation of integrability.

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