A function is continuous at a point if its graph has no break there, meaning the limit exists and equals the function’s value at that point.
Definition
A function f is continuous at x = a if:
- f(a) is defined.
- The limit of f(x) as x approaches a exists.
- That limit equals f(a).
Meaning
You can draw the graph without lifting your pencil.
Types of discontinuity
Jump, removable (hole) and infinite.
Examples
Polynomials are continuous everywhere. 1/x isn’t continuous at 0.
Link
Differentiable implies continuous.
Formal definition
A function f is continuous at x = a if three conditions hold: f(a) is defined, the limit of f(x) as x approaches a exists, and the limit equals f(a). A function is continuous on an interval if it is continuous at every point of that interval. Informally, the graph can be drawn without lifting the pencil.
- Removable discontinuity: a hole, where the limit exists but differs from the value or the value is undefined, as in (x squared minus 1) divided by (x minus 1) at x = 1.
- Jump discontinuity: left and right limits differ, as in the floor function.
- Infinite discontinuity: the function goes to infinity, such as 1 over x at 0.
- Polynomials, sine, cosine and exponential functions are continuous everywhere.
Why it matters
The intermediate value theorem states that a continuous function on [a, b] takes every value between f(a) and f(b). The extreme value theorem says a continuous function on a closed interval has a maximum and minimum. Differentiable functions are always continuous, but not the reverse.
Formal definition
A function f is continuous at a point c if the limit of f(x) as x approaches c equals f(c). It requires the value to exist, the limit to exist and both to match.
Why it matters
Continuous functions satisfy the intermediate value theorem. Types of discontinuity are removable, jump and infinite.
Definition
A function f is continuous at a point a if three conditions hold: f(a) is defined, the limit of f(x) as x approaches a exists, and that limit equals f(a). A function is continuous on an interval if it is continuous at every point in it. Intuitively the graph can be drawn without lifting the pen.
Types of discontinuity and key facts
Discontinuities are removable (a hole), jump (left and right limits differ) or infinite (vertical asymptote). Polynomials, sine, cosine and exponential functions are continuous everywhere, while rational functions are continuous except where the denominator is zero. The Intermediate Value Theorem says a continuous function on a closed interval takes every value between f(a) and f(b), and the Extreme Value Theorem guarantees a maximum and minimum on a closed interval. Differentiable functions are always continuous, but continuous functions need not be differentiable, as with the absolute value at zero.
Based on data from Khan Academy and Paul’s Online Math Notes
