What are piecewise functions?

Question

What are piecewise functions, including those made up of segments and/or other curves, continuous and otherwise?

Answer 1

Piecewise functions are made up of different "pieces." The function as a whole does not follow one rule; each piece follows its own rule. Each rule also has its own constraints for the value(s) of x that satisfy that rule. Here is an example:

f(x) = {

x^2 if x < 0

1 if 0 <= x <= 1

2x if x > 1

This function has three different "pieces." Between negative infinity and x = 0, the function looks like a parabola. Between x = 0 and x = 1, the function looks like a horizontal line. Between x = 1 and positive infinity, the function looks like a line with a slope of 2.

Answer 2

Piecewise functions are not described by the same function all the way along the graph. From zero on left it may be y=x^2, but to the right of zero the function is y=x. The result is a function that can look kind of choppy. If all the pieces of the function touch, then it is continuous. But if one piece ends at one place, and the other piece starts at another place, there will be a break in the line. That function is discontinuous.