Continuity (calculus)
A function is continuous at a point if it is defined there, its limit exists at that point, and the limit equals the function's value. Continuity ensures there are no sudden jumps, holes, or vertical asymptotes in the graph of a function at that point. Continuous functions are fundamental to calculus because key theorems, such as the Intermediate Value Theorem and the Extreme Value Theorem, require continuity on an interval.
Key Formula
lim (x → a) f(x) = f(a)
LaTeX: \lim_{x \to a} f(x) = f(a)
| Symbol | Meaning | Unit |
|---|---|---|
| f(a) | value of the function at point a | dimensionless |
| a | the point of interest | dimensionless |
| lim f(x) | the limit of f as x approaches a | dimensionless |
Worked Example
Problem
Determine whether f(x) = (x² − 9)/(x − 3) is continuous at x = 3. If not, identify the type of discontinuity.
Solution
Step 1: Check if f(3) is defined: f(3) = (9 − 9)/(3 − 3) = 0/0 — undefined. So f is NOT defined at x = 3. Step 2: Find the limit as x → 3: Factor numerator: (x − 3)(x + 3)/(x − 3) = x + 3. Step 3: lim (x → 3) = 3 + 3 = 6. The limit exists but f(3) is not defined. Step 4: This is a removable discontinuity (a hole) at x = 3.
Answer
f(x) has a removable discontinuity at x = 3; the limit is 6 but f(3) is undefined.
Types of Discontinuity in Calculus
| Type | Description | Limit Exists? | f(a) Defined? | Example |
|---|---|---|---|---|
| Removable | Hole in the graph | Yes | No (or ≠ limit) | f(x) = (x²−1)/(x−1) |
| Jump | Left and right limits differ | No | May or may not be | Floor function at integers |
| Infinite | Vertical asymptote | No | No | f(x) = 1/x at x = 0 |
| Oscillating | Function oscillates rapidly | No | May or may not be | sin(1/x) at x = 0 |
Interactive Tools
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Related Terms
Limit (calculus)
A limit describes the value that a function approaches as its input approaches a given point, even if the function is not defined at that point. Limits are the foundational concept of calculus, underpinning the rigorous definitions of derivatives and integrals. They are essential for analysing the behaviour of functions near discontinuities, at infinity, and for understanding rates of change.
Derivative (calculus)
The derivative of a function at a point measures the instantaneous rate of change of the function's output with respect to its input at that point, and geometrically represents the slope of the tangent line to the function's graph. Derivatives are defined as the limit of the difference quotient as the interval shrinks to zero. They are central to physics, engineering, economics, and all sciences wherever rates of change or optimisation are relevant.
Differentiation
Differentiation is the process of computing the derivative of a function, yielding a new function that expresses the rate of change of the original at every point in its domain. It involves applying systematic rules — such as the power rule, product rule, chain rule, and quotient rule — to transform a given function into its derivative. Differentiation is used extensively in physics for velocity and acceleration, in economics for marginal analysis, and in engineering for optimisation and control systems.
From the Latin "continuus" meaning uninterrupted or connected. The rigorous mathematical definition of continuity was formalised by Cauchy and Weierstrass in the 19th century as part of the arithmetisation of analysis.