How do you find discontinuities?
In this video we talk about how to find discontinuities in a function.
0:02 How do you find the discontinuities when you have a picture of the graph of the function? // You need to look for any point where there’s any kind of hole, break, jump, asymptote, or endpoint in the graph. These will all be discontinuities. If you’re tracing the graph from left to right, and you have to pick up your pencil to continue tracing, then there’s a discontinuity at that point. Endpoints are technically discontinuous because there’s only a one-sided limit on one side, which means there’s no general limit, which means the endpoint is technically discontinuous. In order for the function to be discontinuous at an asymptote, the curve needs to exist on both sides of the asymptote. If the curve only exists on one side of the asymptote, then there’s no discontinuity at that point.
1:32 How do you find the discontinuities of a rational function? // Rational functions are fractions with polynomials in the numerator and denominator. Any value that makes the denominator of the fraction 0 is going to produce a discontinuity. If the zero value can be canceled out by factoring, then that value is a point discontinuity, which is also called a removable discontinuity. If the zero value can’t be canceled out by factoring, then that value is an infinite discontinuity, which is also called an essential discontinuity. You can also think of this as just a vertical asymptote for the function.
2:52 How do you find the discontinuities of a piecewise defined function? // You can have discontinuities within each “piece” of the piecewise defined function, just like you could for any other kind of function. But piecewise functions can also be discontinuous at the “break point”, which is the point where one piece stops defining the function, and the other one starts. If the two pieces don’t meet at the same value at the “break point”, then there will be a jump discontinuity at that point. For that reason, jump discontinuities are really common for piecewise functions.