High School Continuity and Discontinuity Tester
Analyze a function f(x) at a specific point c to determine if it is continuous.
Try: (x^2-4)/(x-2), 1/x, abs(x)/x, or sin(x)/x
Conclusion
The 3-Step Continuity Test
Function Behavior
About the Continuity and Discontinuity Tester
In calculus, continuity is a core concept that describes functions without breaks, jumps, or holes. Intuitively, a function is continuous if you can draw its graph without lifting your pencil from the paper. To define this mathematically, we use limits.
The 3-Step Continuity Test
A function f(x) is continuous at a point x = c if and only if all three of these conditions are met:
- f(c) is defined: The function must have a real value at c.
- The limit exists: As x approaches c from the left and right, f(x) must approach the same finite value.
- Limit equals Value: The value of the limit must equal the function value f(c).
Types of Discontinuity
If any condition fails, the function is discontinuous. There are three main types:
- Removable (Hole): The limit exists, but f(c) is undefined or not equal to the limit. This often happens in rational functions where a factor cancels out.
- Jump: The left-hand and right-hand limits exist but are not equal. Common in piecewise functions.
- Infinite: The function approaches infinity (vertical asymptote) from one or both sides.
Why use a Continuity and Discontinuity Tester?
Calculating limits numerically and verifying definitions can be tedious. This tool automates the “Left vs. Right” check and visualizes the specific behavior at the point of interest, making abstract calculus concepts concrete.
FAQ
Yes, this tool is completely free for educational use.
It’s a point where the graph has a “hole”. You could make the function continuous just by redefining the value at that single point.
It handles functions like abs(x)/x which behave like piecewise functions. For explicit multi-line piecewise definitions, standard math syntax is required.
If the function oscillates wildly (like sin(1/x) near 0) or is undefined in a neighborhood, the limit may not exist.