#116 part b We have =0, but lim4:38A inst-fs-iad-prod.inscloudgate.netNote that Theorem3 of ChapterI will probably help here.]116 of 220Jequences and ContinuityThere is an alternative way to prove that the functionD(z) = , if z is rational(0, if r is irrationalis not continuous at a + 0. We will examine this by looking at the relationshipbetween our definitions of convergence and continuity. The two ideas are actu-ally quite closely connected, as illustrated by the following very useful theorem.Theorem 15. The function f is continuous at a if and only if ƒ satisfies thefollowing property:V sequences (r,,), if lim r, = a then lim f(r,) = f(a).nTheorem 15 says that in order for f to be continuous, it is necessary andsufficient that any sequence (zn) converging to a must force the sequence (f(zn))to converge to f(a). A picture of this situation is below though, as always, theformal proof will not rely on the diagram.CONTINUITY: WHAT IT ISN'T AND WHAT IT Is111This theorem is especially useful for showing that a function f is not con-tinuous at a point a; all we need to do is exhibit a sequence (r,) oconvergingto a such that the sequence lim, f(rn) does not converge to f(a). Let'sdemonstrate this idea before we tackle the proof of Theorem 15,Example 12. Use Theorem 15 to prove that(z) =10,if z=0is not continuous at 0.Proof: First notice that f can be written asif r0S(z) = {-1 if x 0. Use120 of 220 on with e = 20. The picture is below; make it formal.g(a)g(a) – 9(@a-8a+8For the case g(a)

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#116 part b We have =0, but lim4:38A inst-fs-iad-prod.inscloudgate.netNote that Theorem3 of ChapterI will probably help here.]116 of 220Jequences and ContinuityThere is an alternative way to prove that the functionD(z) = , if z is rational(0, if r is irrationalis not continuous at a + 0. We will examine this by looking at the relationshipbetween our definitions of convergence and continuity. The two ideas are actu-ally quite closely connected, as illustrated by the following very useful theorem.Theorem 15. The function f is continuous at a if and only if ƒ satisfies thefollowing property:V sequences (r,,), if lim r, = a then lim f(r,) = f(a).nTheorem 15 says that in order for f to be continuous, it is necessary andsufficient that any sequence (zn) converging to a must force the sequence (f(zn))to converge to f(a). A picture of this situation is below though, as always, theformal proof will not rely on the diagram.CONTINUITY: WHAT IT ISN'T AND WHAT IT Is111This theorem is especially useful for showing that a function f is not con-tinuous at a point a; all we need to do is exhibit a sequence (r,) oconvergingto a such that the sequence lim, f(rn) does not converge to f(a). Let'sdemonstrate this idea before we tackle the proof of Theorem 15,Example 12. Use Theorem 15 to prove that(z) =10,if z=0is not continuous at 0.Proof: First notice that f can be written asif r>0S(z) = {-1 if x <0if x = 0To show that f is not continuous at 0, all we need to do is create a singlesequence (r,) which converges to 0, but for which the sequence (f (xn)) does notconverge to f(0) = 0. For a function like this one, just about any sequence willdo, but let's use (1), just because it is an old familiar friend.()-We have lim-= 0, but lim f= lim 1=1#0 = f(0). Thus byTheorem 15 f is not continuous at 0. 4:37A inst-fs-iad-prod.inscloudgate.neta yo trreeere o er y TTthat g(z) +0 for all r € (a – 6, a + 6).Problem 114. Prove Lemma (Hint: Consider the case where g(a) > 0. Use120 of 220 on with e = 20. The picture is below; make it formal.g(a)g(a) – 9(@a-8a+8For the case g(a) <0, consider the function -g.)A consequence of this lemma is that if we start with a sequence (r,) con-verging to a, then for n sufficiently large, g(r.) #0.Problem 115. Use Theorem 15 to prove that if f and g are continuous at aand g(a) # 0, then f/g is continuous at a.Theorem 17. Suppose f is continuous at a and g is continuous at f(a). Thengof is continuous at a. [Note that (go f)(x) = g(f(x)).)Problem 116. Prove Theorem 17(a) Using the definition of continuity.(b) Using Theorem 15,The above theorems allow us to build continuous functions from other con-tinuous functions. For example, knowing that f(r) = r and g(x) = c arecontinuous, we can conclude that any polynomial,p(x) = anr" + an-12"-1 + ... + a,x+ aoCONTINUITY: WHAT IT ISN'T AND WHAT IT Is115is continuous as well. We also know that functions such as f(z) = sin (e*) arecontinuous without having to rely on the definition.Problem 117. Show that each of the following is a continuous function at everypoint in its domain.1. Any polynomial.2. Any rational function. (A rational function is defined to be a ratio ofpolynomials.)3. cos z.4. The other trig functions: tan(r), cot(r), sec(r), and cse(r).Problem 118. What allows us to conclude that f(r) = sin (e) is continuousat any point a without referring back to the definition of continuity?Theorem [15 can also be used to study the convergence of sequences. Forexample, since f(z) = e is continuous at any point and lim,nx "1 = 1,then lim. e() = e. This also illustrates a certain way of thinking aboutcontinuous functions. They are the ones where we can "commute" the functionand a limit of a sequence. Specilfically, if } is continuous at a and lim, In = a,

Accepted Answer

It is given that, f is continuous at a and g is continuous at f(a).
We have to show that gof is…

Theorem 17. Suppose f is continuous at a and g is continuous at f(a). Then gof is continuous at a. [Note that (go f)(x) = g(f(x))./ Problem 116. Prove Theorem 17 (a) Using the definition of continuity. (b) Using Theorem 15, The obe build tinuo funotione from