For , let , where and are relatively prime integers, then which one of the following…

MATHEMATICS · Single correct · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

For

x > 0

, let

h(x) = \begin{cases} \frac{1}{q}, & \text{if } x = \frac{p}{q} \\ 0, & \text{if } x \text{ is irrational} \end{cases}

, where

p

and

q > 0

are relatively prime integers, then which one of the following does not hold good? (1)

h(x)

is discontinuous for all

x

(0, \infty)

(2)

h(x)

is continuous for each irrational in

(0, \infty)

(3)

h(x)

is discontinuous for each rational in

(0, \infty)

(4)

h(x)

is not derivable for all

x

(0, \infty)