Suppose S is defined on an open interval containing x=a except possibly at x=a itself. Its derivative S′(x) is known to possess a removable discontinuity at x=a. A particular solution S is specified by the condition S(b)=c for some b=a.
(a) Explain why S must be continuous at x=a, even though S′(x) is not defined there. Base your reasoning on appropriate fundamental principles.
(b) Describe what additional information, if any, would be required to conclude that S is differentiable at x=a. Provide a clear, logically ordered argument.