### 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.

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

 { ex 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.

 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

See Explanation.

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