The function h(x)=x3−3x is continuous on 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.