A continuity, with parameter, example

Consider the function

This actually defines an infinite number of functions, one for each value of r. Depending on the value of r, the function may be continuous or discontinuous at x=0.

Note that each of the two parts of the definition is a function that is continuous for all x. So the only place f(x) could be discontinuous is at x=0.

You should show that the limit of f(x), as x approaches 0 from the left is 2r²-1 and the limit of as x approaches 0 from the right is r.

Also, f(0)=r, so f(x) is continuous at 0 (and hence everywhere) only if r = 2r²-1. Solving this equation, you find that f(x) is continuous at 0 only if r=1 or r=-1/2.

Below is an animation showing the graph of f(x) for various value of r. The parts of the graph to the left and right of zero are in two different colors to emphasize the difference in their definitions. As you can see, most of the time there is a jump in the graph at 0. For only two values of r do the left and right parts of the graph meet "continuously" at 0: r=-1/2 and r=1.