Discontinuous function
Classify and graph discontinuous functions: identify jump discontinuities, infinite discontinuities (vertical asymptotes), and removable discontinuities (holes) from the function's equation.
Key Concepts
A discontinuous function is any function whose graph has gaps, jumps, or asymptotes, making it impossible to draw without lifting your pencil. These breaks can appear as holes, called points of discontinuity, or as vertical lines the graph approaches but never touches, known as asymptotes. It's essentially a graph with one or more interruptions in its path.
A function with a hole, like $f(x) = \frac{x^2 4}{x 2}$, is discontinuous at $x = 2$. A function with a vertical asymptote, such as $g(x) = \frac{1}{x}$, is discontinuous at $x = 0$. A step function, which models costs like parking fees that jump at certain times, is discontinuous.
Think of a road with a broken bridge or a sudden sinkhole. You can't drive straight through! A discontinuous function's graph is just like that—it has breaks, forcing you to 'lift your pencil' to get from one part to the next.
Common Questions
What are the three types of discontinuity in a function?
Jump discontinuity: the left and right limits exist but differ, creating an abrupt jump. Infinite discontinuity: the function approaches positive or negative infinity near a point, appearing as a vertical asymptote. Removable discontinuity: a hole where the limit exists but the function value is missing or different.
How do you identify a removable discontinuity (hole) in a rational function?
Factor both the numerator and denominator. If a factor cancels completely from both, the corresponding x-value is a removable discontinuity (hole), not a vertical asymptote. The graph has a hole at that x-value with the y-value given by the simplified expression.
What is the difference between a hole and a vertical asymptote on a graph?
A hole occurs when a common factor cancels, leaving a single missing point. A vertical asymptote occurs when the denominator factor does not cancel, causing the function to grow without bound. Both are undefined at the restricted x-value, but their graphical behavior is completely different.