Consider the following four statements about a function F defined on an open interval containing x=c. For each statement, indicate whether it is always true, sometimes true, or never true. In every case, provide a brief justification rooted in limits or specific examples.
I. If F is differentiable at x=c, then F is continuous at x=c.
II. If F is continuous at x=c, then F is differentiable at x=c.
III. If F is not continuous at x=c, then F is not differentiable at x=c.
IV. If F′(c) does not exist, then x→climF(x) does not exist.
a. Analyze statement I and classify it as always, sometimes, or never true. Supply a concise justification.
b. Analyze statement II and classify it as always, sometimes, or never true. Supply a concise justification.
c. Analyze statement III and classify it as always, sometimes, or never true. Supply a concise justification.
d. Analyze statement IV and classify it as always, sometimes, or never true. Supply a concise justification.