The function is continuous on

MATHEMATICS · Single correct · A Level · Mathematics · Algebra and Functions · Algebra and Functions

Problem

The function

h(x)=x^{3}-3x

is continuous on

\mathbb{R}

, yet its inverse function

h^{-1}

is not continuous when defined on the entire range of

h

. Which of the following best explains why

h^{-1}

fails to be continuous? (a)

h

has vertical asymptotes, which translate into jumps in

h^{-1}

. (b)

h

is not one-to-one because it changes from increasing to decreasing, so a single

y

-value corresponds to multiple

x

-values. (c)

h

has critical points where

h'(x)=0

, giving vertical tangent lines in

h^{-1}

that break continuity. (d)

h

is unbounded below, preventing its inverse from being continuous on the whole range.