Continuity: Step-by-Step Guide
A function f is continuous at x = a when three conditions all hold: f(a) is defined, the limit of f(x) as x approaches a exists, and that limit equals f(a). If any condition fails, f has a discontinuity (removable, jump, or infinite) at a.
Math Racer skill guide · Calculus 1 · AP Calc AB / College
How to solve it, step by step
- Check that f(a) is defined. Substitute x = a into the function. If you get a real number, this condition is met. If you get 0/0, division by zero, a square root of a negative, or a log of zero or negative, the function is not defined at a.\[f(a) \text{ must be a real number}\]
- Compute the two-sided limit at a. Find the left- and right-hand limits. If they are equal and finite, the limit exists. If they differ, you have a jump discontinuity. If either is infinite, you have an infinite discontinuity.\[\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L\]
- Compare the limit to f(a). If L = f(a), f is continuous at a. If L exists but does not equal f(a) (or f(a) is undefined), the discontinuity is removable.\[\lim_{x \to a} f(x) = f(a)\]
- Classify any discontinuity. Removable: limit exists but disagrees with f(a). Jump: one-sided limits disagree. Infinite: a one-sided limit is plus or minus infinity (vertical asymptote).\[\text{Removable, Jump, or Infinite}\]
Worked example
Determine whether f(x) = (x^2 - 4)/(x - 2) is continuous at x = 2; if not, classify the discontinuity.
Step 1: Check f(2). Substituting gives (4 - 4)/(2 - 2) = 0/0, which is undefined, so condition 1 fails.
Step 2: Factor: (x^2 - 4)/(x - 2) = ((x-2)(x+2))/(x-2) = x + 2 for x not equal to 2.
Step 3: Take the limit: lim x->2 (x + 2) = 4. The limit exists and equals 4.
Step 4: Since the limit exists but f(2) is undefined, the discontinuity is removable.
Step 5: We could repair it by defining f(2) = 4, making the new function continuous everywhere.
Verification: f(1.99) = 3.99 and f(2.01) = 4.01, confirming the limit value of 4.
Tips and common mistakes
- Always test all three continuity conditions; missing one is the most common error.
- Polynomials, sine, cosine, and exponentials are continuous everywhere; rational functions are continuous wherever the denominator is nonzero.
- If direct substitution gives 0/0, try factoring or rationalizing — this often reveals a removable discontinuity.
- Jump discontinuities most often appear at the boundary x-values of piecewise functions.
- An infinite discontinuity (vertical asymptote) means the limit is not a finite number, so f cannot be continuous there.
- For piecewise functions, set the two pieces equal at the boundary to find parameters that make f continuous.
- Continuity on a closed interval [a, b] requires interior continuity plus one-sided continuity at the endpoints.
- Differentiability implies continuity, but not the reverse — for example, |x| is continuous but not differentiable at 0.