Differentiation rule for piecewise definition by interval

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Statement

Everywhere version

Suppose f_1 and f_2 are functions of one variable, such that both of the functions are defined and differentiable everywhere. Consider a function f, defined as follows:

f(x) := \left\lbrace \begin{array}{rl} f_1(x), &  x < c \\ f_2(x), & c < x \le a_2 \\v, & x = c \end{array}\right.

Then, we have the following for continuity:

We have the following for differentiability:

  • f is left differentiable at c iff v = f_1(c), and in this case, the left hand derivative equals f_1'(c).
  • f is right differentiable at c iff v = f_2(c), and in this case, the right hand derivative equals f_2'(c).
  • f is differentiable at c iff (v = f_1(c) = f_2(c) and f_1'(c) = f_2'(c)), and in this case, the derivative equals the equal values f_1'(c) and f_2'(c).

Piecewise definition of derivative

If the conditions for differentiability at c are violated, we get the following piecewise definition for f', which excludes the point c from its domain:

f'(x) := \left \lbrace \begin{array}{rl} f_1'(x), & x < c \\ f_2'(x), & x > c \\\end{array}\right.

If the conditions for differentiability at c are satisfied, we get the following piecewise definition for f', which includes the point c in its domain:

f'(x) := \left \lbrace \begin{array}{rl} f_1'(x), & x < c \\ f_2'(x), & x > c \\ u, & x = c\end{array}\right.

where u = f_1'(c) = f_2'(c). In particular, the value at c can be included in either the left side or the right side definition.

Version for higher derivatives

Suppose f_1 and f_2 are functions of one variable, such that both of the functions are defined and k times differentiable everywhere (and hence in particular the functions and their first k-1 derivatives are continuous), for some positive integer k. Consider the function:

f(x) := \left\lbrace \begin{array}{rl} f_1(x), &  x < c \\ f_2(x), & c < x \le a_2 \\v, & x = c \end{array}\right.

Then, f is k times differentiable at c if we have all these conditions: f_1(c) = f_2(c) = v, f_1'(c) = f_2'(c), <math>\dots, f_1^{(k)}(c) = f_2^{(k)}(c). In other words, the values should match, and the values of each of the derivatives up to the k^{th} derivative should match. In that case, the k^{th} derivative of f at math>c</math> equals the equal values f_1^{(k)}(c) = f_2^{(k)}(c).

The general piecewise definition of f^{(k)} is, in this case:

f^{(k)}(x) := \left \lbrace \begin{array}{rl} f_1^{(k)}(x), & x < c \\ f_2^{(k)}(x), & x > c \\ u_k, & x = c\end{array}\right.

where u_k = f_1^{(k)}(c) = f_2^{(k)}(c).

Local generalization

The above holds with the following modification: we only require f to be defined as f_1 on the immediate left of c (i.e., on some interval of the form (c - \delta,c) for \delta > 0 and as f_2 on the immediate right of c (i.e., on some interval of the form (c,c + \delta) for \delta > 0). Further, we only require that f_1 and f_2 be defined and differentiable on open intervals containing c, not necessarily on all of \R.