Let be the -periodic extension to of the function defined on the interval by

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Let

p

be the

2

-periodic extension to

\mathbb{R}

of the function defined on the interval

[-1,1]

p(x)=|x|

. In other words,

p(x+2)=p(x)

for all

x

, and

p(x)=|x|

[-1,1]

. (a) Prove that

p

is continuous on

\mathbb{R}

. (b) Identify all real numbers

c

at which

p

fails to be differentiable. Provide a rigorous justification that directly evaluates the one-sided limits of the difference quotient at each such

c

. (c) For a point

a

at which

p

is differentiable, compute

p'(a)

in closed form. Express your answer as a function of

a

defined over the appropriate domain.