structural:summarization, linguistic:paraphrasing
Continuity and differentiability of piecewise-defined functions

A piecewise-defined function is continuous on a given interval in its domain if the following conditions are met:

The pictured function, for example, is piecewise-continuous throughout its subdomains, but is not continuous on the entire domain, as it contains a jump discontinuity at x 0 {\displaystyle x_{0}}. The filled circle indicates that the value of the right sub-function is used in this position.

For a piecewise-defined function to be differentiable on a given interval in its domain, the following conditions have to fulfilled in addition to those for continuity above:

Some sources only examine the function definition, while others acknowledge the property iff the function admits a partition into a piecewise definition that meets the conditions.