Analysis Suppose f (x) is defned on (a, b) and continuous at c, where a < c < b. (a) Prove that if |f | is differentiabl
Posted: Thu May 12, 2022 1:42 pm
Analysis
Suppose f (x) is defned on (a, b) and continuous at c, where a
< c < b.
(a) Prove that if |f | is differentiable at c, then f is
differentiable at c. (Hint: Consider the cases f (c) > 0, f (c)
< 0, and f (c) = 0 separately.)
(b) Is the condition that f is continuous at c necessary in (a)?
Justify your answer.
Suppose f (x) is defned on (a, b) and continuous at c, where a
< c < b.
(a) Prove that if |f | is differentiable at c, then f is
differentiable at c. (Hint: Consider the cases f (c) > 0, f (c)
< 0, and f (c) = 0 separately.)
(b) Is the condition that f is continuous at c necessary in (a)?
Justify your answer.