Differentiation rule for piecewise definition by interval
Suppose and are functions of one variable, such that both of the functions are defined and differentiable everywhere. Consider a function , defined as follows:
Then, we have the following for continuity:
- The left hand limit of at equals .
- The right hand limit of at equals .
- is left continuous at iff .
- is right continuous at iff .
- is continuous at iff .
We have the following for differentiability:
- is left differentiable at iff , and in this case, the left hand derivative equals .
- is right differentiable at iff , and in this case, the right hand derivative equals .
- is differentiable at iff ( and ), and in this case, the derivative equals the equal values and .
Piecewise definition of derivative
If the conditions for differentiability at are violated, we get the following piecewise definition for , which excludes the point from its domain:
If the conditions for differentiability at are satisfied, we get the following piecewise definition for , which includes the point in its domain:
where . In particular, the value at can be included in either the left side or the right side definition.
Version for higher derivatives
Suppose and are functions of one variable, such that both of the functions are defined and times differentiable everywhere (and hence in particular the functions and their first derivatives are continuous), for some positive integer . Consider the function:
Then, is times differentiable at if we have all these conditions: , , . In other words, the values should match, and the values of each of the derivatives up to the derivative should match. In that case, the derivative of at math>c</math> equals the equal values .
The general piecewise definition of is, in this case:
The above holds with the following modification: we only require to be defined as on the immediate left of (i.e., on some interval of the form for and as on the immediate right of (i.e., on some interval of the form for ). Further, we only require that and be defined and differentiable on open intervals containing , not necessarily on all of .