Piecewise Functions

The continuity of piecewise functions must be handled on a case-by-case basis. It is necessary to look separately

On each "piece". If f(x) is defined to be the function g(x) on some interval, then f(x) and g(x) have the same continuity properties except at the endpoints of the interval.
At the endpoints, where two "pieces" come together. The piecewise function f(x) is continuous at such a point if and only of the left- and right-hand limits of the pieces agree and are equal to the value of the f.

A graph often helps determine continuity of piecewise functions, but we should still examine the algebraic representation to verify graphical evidence.

Graph of f(x)

Example: The function

	{	x²	x < -1
f(x) =		x	-1 x < 1
		-cos(x)	x 1

appears in the Field Guide section on piecewise functions. It is continuous at all points with the exception of x=-1, where the "pieces" do not fit smoothly together. Note that f(x) is continuous at x=1 because

lim x =	lim cos(x)	= 1 = f(1).
x --> 1^-	x --> 1⁺

Equipment Check: Let g(x) be the piecewise function

	{	e^x	x < 0
g(x) =		x + 1	0 x 2
		ln(x)	x > 2

Decide whether g(x) is continuous at the points x=0 and x=2.

See Explanation.

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g(x) is continuous at x=0 g(x) is discontinuous at x=0
g(x) is continuous at x=2 g(x) is discontinuous at x=2