If, is continuous, and, what is the value of.
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(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.
[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 ยท 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.
If f is a continuous function, what is the limit as of the average value of f on the interval.
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is continuous, and
, what is the value of
., is continuous, and, what is the value of.
, is continuous, and, what is the value of.
is continuous everywhere.
is continuous, does it follow that f is continuous? If yes, prove it. If not, find a counter example.
of the average value of f on the interval
.