Continuity in one sentence
A function f is continuous at x = a when its graph has no break there. Precisely, three things must hold: f(a) is defined, the limit of f(x) as x approaches a exists, and the two are equal. If any one fails, f is discontinuous at a. Polynomials, sin, cos and e^x are continuous everywhere, and a rational function is continuous everywhere its denominator is not zero.
The limit laws
If the limits of f and g exist as x approaches a, then limits respect arithmetic. The limit of a sum is the sum of the limits, and the same for differences and constant multiples. The limit of a product is the product of the limits. The limit of a quotient is the quotient of the limits as long as the bottom limit is not zero. A root or a power passes straight through as well. For example, if f approaches 4 and g approaches 3, then 2f + g approaches 11 and f/g approaches 4/3.
For a continuous function you can simply substitute: the limit of a polynomial or other continuous function at a point equals its value there.
Types of discontinuity
- Removable: a hole, where the limit exists but f(a) is missing or wrong. Redefining f(a) repairs it.
- Jump: the one-sided limits exist but differ.
- Infinite: the function grows without bound near a, giving a vertical asymptote.
Intermediate value theorem
If f is continuous on a closed interval [a, b] and N is any value between f(a) and f(b), then f(c) = N for some c in the interval. In particular, a sign change from negative to positive guarantees a root. The theorem needs continuity; a jump can skip values.
Worked example
Choose k so that f(x) = kx + 2 for x < 3 and f(x) = x^2 - 1 for x >= 3 is continuous at 3.
Right side: f(3) = 9 - 1 = 8. Left limit: 3k + 2. Setting 3k + 2 = 8 gives k = 2. With k = 2, both sides equal 8 and the graph joins smoothly.
Second example: show that g(x) = x^3 + x - 3 has a root between 1 and 2. Here g(1) = -1 and g(2) = 7. Since g is a polynomial it is continuous, and 0 lies between -1 and 7, so some c between 1 and 2 has g(c) = 0.
Squeeze theorem
If h(x) is trapped between two functions that approach the same limit L at a, then h(x) approaches L too.
Common mistakes
- Saying the limit of a quotient is the quotient of limits when the bottom limit is 0.
- Applying the intermediate value theorem to a function that is not continuous on the whole interval.
- Forgetting to check all three conditions of continuity.