Continuity at a point describes when a function has no break, hole, or jump at a specific input value. Visually, the graph passes smoothly through the point without needing to lift your pencil. This idea matters because many calculus tools, including limits, derivatives, and the Intermediate Value Theorem, rely on functions behaving predictably.
A continuous point connects the formula, the graph, and the limit into one consistent value.
For a function f to be continuous at x = a, three conditions must all be true: f(a) is defined, the limit as x approaches a exists, and the limit equals f(a). If even one condition fails, the function is not continuous at that point. Continuity on an interval means the function is continuous at every point in that interval, with one-sided continuity at endpoints when needed.
Common discontinuities include holes, jumps, and vertical asymptotes, each caused by a different failure of the continuity conditions.
Key Facts
- Continuity at x = a means lim x->a f(x) = f(a).
- Condition 1: f(a) is defined, so the function has an actual value at x = a.
- Condition 2: lim x->a f(x) exists, so the left-hand and right-hand limits are equal.
- Condition 3: lim x->a f(x) = f(a), so the approaching value matches the function value.
- A function is continuous on an interval if it is continuous at every point in that interval.
- For endpoints, use one-sided continuity: lim x->a+ f(x) = f(a) or lim x->b- f(x) = f(b).
Vocabulary
- Continuity at a point
- A function is continuous at x = a when its value and its limiting value at that point are the same.
- Limit
- A limit is the value a function approaches as the input gets close to a given number.
- Function value
- The function value f(a) is the actual output of the function when the input is x = a.
- One-sided limit
- A one-sided limit is the value a function approaches from only the left or only the right side of a point.
- Discontinuity
- A discontinuity is a point where a function fails to be continuous because of a hole, jump, asymptote, or mismatch.
Common Mistakes to Avoid
- Checking only whether f(a) is defined. This is wrong because a defined point can still be discontinuous if the limit does not exist or does not equal f(a).
- Assuming a graph is continuous because the left side looks smooth. This is wrong because continuity at a point requires matching behavior from both the left and the right.
- Forgetting to compare the limit with the actual function value. This is wrong because a removable discontinuity can have a limit but still fail continuity if f(a) is missing or different.
- Using a two-sided limit at an endpoint of a closed interval. This is wrong because endpoints require one-sided continuity from inside the interval.
Practice Questions
- 1 Let f(x) = x^2 + 3x. Determine whether f is continuous at x = 2 by comparing f(2) and lim x->2 f(x).
- 2 Let f(x) = (x^2 - 9)/(x - 3) for x != 3, and f(3) = 5. Is f continuous at x = 3? Show the limit and compare it with f(3).
- 3 A graph has an open circle at (2, 4) and a filled dot at (2, 1), while the curve approaches y = 4 from both sides. Explain which continuity condition fails at x = 2 and why.