(a) Show that the absolute value function is continuous everywhere.
(b) Prove that if f is a continuous function on an interval, then so is
(c) Is the converse of the statement in part (b) also true? In other words, if is continuous, does it follow that f is continuous? If yes, prove it. If not, find a counter example.

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Elin HesselLv2

28 May 2020

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[8] 9. Either prove the statement is true or show that it is false. (a) sin (sin(2)) = 27. Solution: It is FALSE. The range of sin-' r is (-1,1], hence sin (sin(27)) = sin (0) = 0 (b) If f is continuous at a, then f is continuous at a. Solution: It is TRUE. Let g(x) = \f()]. If lim f(x) = f(a), then we have since the absolute value function is continuous lim g(x) = lim f() = lim f(x) = f(a) = g(a) So, g = \f is continuous at a. (c) If [g] is continuous at a, then g is continuous at a. -1 if r <0 Solution: It is FALSE. Let g(x) = 3 · Then, g = 1, so g is continuous at 0 i if >0 by the Continuity Theorems, but lim g(t) = -1 + lim g(1) 240- 1- 0+ So, g(x) is not continuous at a. (d) If f is continuous at a, then f is differentiable at a. Solution: It is FALSE. We know that f(1) = 2 is continuous at 0, but is not differentiable at 0.

A fixed point of a function f is a number c in its domain such that . (The function doesn’t move c; it stays fixed.)

(a) Sketch the graph of a continuous function with domain whose range also lies in. Locate a fixed point of f.

(b) Try to draw the graph of a continuous function with domain and range in that does not have a fixed point. What is the obstacle?

(c) Use the Intermediate Value Theorem to prove that any continuous function with the domain and range in must have a fixed point.

tancoyote204

5. (3 points) Which of the following statements best describes the relationship between continuity and differentiability for a function? Choose only one, and assume that the functions are defined for all real numbers. (A) A function must be both differentiable and continuous at r = a or neither; you cannot have one without the other. (B) If a function is differentiable at I = a then it must also be continuous at r = a. (C) If a function is continuous at r= a then it must also be differentiable at r=a. (D) There is at least one function that is continuous for some r = a where it is not differentiable, and at least one other function that is differentiable at some r = a where it is not continuous. (E) Two of the above statements mean the same thing, and are both the correct choice.

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