Recall that N = {1, 2, 3, ...} is the set of natural numbers. Define the set R to be the set of their reciprocals, i.e.

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Recall That N 1 2 3 Is The Set Of Natural Numbers Define The Set R To Be The Set Of Their Reciprocals I E 1 (27.87 KiB) Viewed 44 times

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Recall that N = {1, 2, 3, ...} is the set of natural numbers. Define the set R to be the set of their reciprocals, i.e. R = { n = N} = {1, ½, ½, ½ ‚ ... }. 2 3 4 Now consider the following function: ƒ(²) = { if z This is its graph: xif x ER 0 if x R 0.5+

-0.5 This function is discontinuous at every x E R. What type of discontinuities are they? oscillatory Ojump infinite 0.5 O removable

f(x) Does lim fx x-0 lim fx x-0 (₂) DNE because f(0) is undefined. 1(x) lim fx x-0 exist, and why? lim fx x-0 (2) -|x ≤ f(x) ≤ x for all real numbers . 1(₂) exists and equals 0 because the values in R are all positive. exists and equals O by the Squeeze Theorem since lim f x-0 intervals containing 0. DNE because there are infinitely many discontinuities in all open

Is f(x) continuous at x = 0, and why? No, because lim fx 1 (2) x→0 O No, because there are infinitely many discontinuities in all open intervals containing = 0. DNE. Yes, because lim fx x-0 * (*) = No, because f(0) is undefined. Yes, because f(0) = 0. Yes, because lim f(x x-0 = 0. 1(x)=0-s(0).