WHEN is a PIECEWISE defined function continuous/differentiable? | Calculus Tutorial

We explain how to test continuity and differentiability at a point where different pieces of a piecewise-defined function merge. In particular, we study the left-hand limit and the right-hand limit, and the left-hand derivative and the right-hand derivative. In the example in the thumbnail, we use the continuity and the differentiability at x = 1 separately in order to set up two simultaneous equations for a and b. Finally, we solve the simultaneous equations to determine the unique values of a and b such that the function is differentiable (at x = 1, and in general).

The techniques in this video generally illustrate how to determine continuity and differentiability of a function at a point. A mastery of this concept is central to problems that arise in homework, assignments, tests and exams in high school and college calculus.

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