(a) Define continuity of a real-valued function at a point using the limit concept.…
MATHEMATICS · Subjective · AP · Calculus AB · Limits and Continuity · Limits and Continuity
Problem
(a) Define continuity of a real-valued function at a point using the limit concept.
(b) Describe the following three kinds of discontinuities that may occur in the graph of a real function:
(i) removable (point) discontinuity,
(ii) jump discontinuity,
(iii) infinite discontinuity.
For each kind,
(1) draw a small illustrative sketch in your answer book, and
(2) write one algebraic example of a function that exhibits that discontinuity at a specified point, clearly mentioning the point involved.
(c) Explain why every removable discontinuity can be eliminated by redefining the function at a single point, whereas jump and infinite discontinuities cannot be removed in this way.