Key takeaway: Continuity at x = a means three things line up: the function has a value there, the neighbors point to a value, and those two values agree.
Grade 12 – AP Calculus ABAP Calculus AB Unit 1Calculus
Definition of Continuity
Continuity is the mathematical version of drawing a graph without lifting your pencil. A function passes at a point when nothing goes wrong there.
When a test fails, the graph has a discontinuity. A removable one is a single missing or misplaced point, which could be repaired by redefining one value. A jump has two different one-sided values. An infinite one has a vertical asymptote. Polynomials are continuous everywhere, and rational functions are continuous wherever their denominators are nonzero.
Formula and notation
Test \(f\) at \(x=a\) with three checkpoints, all of which must pass:
\[\text{(1) } f(a)\text{ exists},\qquad \text{(2) } \lim_{x\to a}f(x)\text{ exists},\qquad \text{(3) } \lim_{x\to a}f(x)=f(a).\]
Continuity on an interval means the test passes at every point inside it.
Key facts
- Failing any one of the three parts is enough to make the point discontinuous.
- A removable gap can be repaired by assigning the value the neighbors point to.
- Sums, products and compositions of continuous functions are continuous.
- Continuous on a closed interval guarantees the intermediate value property.
Continuity example
Let \(f(x)=\dfrac{x^2+x-12}{x-3}\) for \(x\neq 3\), and \(f(3)=2\). Is \(f\) continuous at \(x=3\)? If not, what value of \(f(3)\) would repair it?
-
Part 1: \(f(3)=2\) is defined.
-
Part 2: factor the top as \((x+4)(x-3)\) and cancel; the quotient equals \(x+4\) for every \(x\neq 3\), so the outputs approach 7.
-
Part 3: the neighbors point to 7 but the function value is 2, and \(7\neq 2\). The test fails.
The function is not continuous at 3. Redefining \(f(3)=7\) removes the gap.
Common mistake
Watch out
Isaac says f is continuous at 3, since a value of 7 exists for the limit.
Why it is wrong
Existence of the limit is only part 2. Continuity also needs the function value to match it. Here \(f(3)=2\), not 7, so the third part fails.
Related terms and lessons
Browse the glossary: A–Z list · AP Calculus AB vocabulary · Calculus