8.6 Continuity

Definition of continuity

Continuity describes functions that behave predictably: no sudden jumps, no holes, no blowing up to infinity. It's one of the most important ideas in analysis because it guarantees that functions behave "nicely" enough for the major theorems of calculus to work.

The intuitive picture is simple: a continuous function is one you can draw without lifting your pencil from the paper. Small changes in input always produce small changes in output. But to do real mathematics with this idea, you need a precise definition.

Formal definition

A function $f(x)$ is continuous at a point $a$ if three conditions are all satisfied:

1. $f(a)$ is defined (the function actually has a value at that point)
2. $\lim_{x \to a} f(x)$ exists (the function approaches some definite value)
3. $\lim_{x \to a} f(x) = f(a)$ (the value it approaches matches the value it actually takes)

If even one of these fails, the function is discontinuous at $a$. A function is continuous on an interval when it's continuous at every point in that interval.

Epsilon-delta definition

The epsilon-delta formulation makes the intuition "small input changes give small output changes" completely rigorous:

Here's how to read that:

• $\epsilon$ represents how much wiggle room you'll tolerate in the output.
• $\delta$ represents how close the input needs to be to $a$ to stay within that tolerance.
• The definition says: no matter how tight a tolerance someone demands on the output, you can always find a close-enough neighborhood around $a$ on the input side to meet it.

This definition forms the backbone of rigorous proofs throughout analysis.

Types of continuity

Different "strengths" of continuity exist, and they matter because stronger forms guarantee more useful properties.

Pointwise continuity

This is the standard definition described above, applied one point at a time. A function can be continuous at some points and discontinuous at others. The key detail: the $\delta$ you find is allowed to depend on both $\epsilon$ and the specific point $a$.

Uniform continuity

Uniform continuity is a stronger requirement. A single $\delta$ must work for all points in the domain simultaneously, not just for one particular point. This means the function's behavior is consistently well-controlled everywhere, with no regions where it changes more and more steeply.

Every uniformly continuous function is pointwise continuous, but not vice versa. For example, $f(x) = x^2$ is continuous everywhere on $\mathbb{R}$ but not uniformly continuous on $\mathbb{R}$, because as $x$ gets large, the function changes faster and faster. However, on any closed bounded interval $[a,b]$, a continuous function is always uniformly continuous (this is a theorem, not obvious).

Absolute continuity

This is stronger still than uniform continuity. A function is absolutely continuous if, roughly speaking, whenever you take a collection of small intervals whose total length is tiny, the total variation of the function over those intervals is also tiny. Absolute continuity guarantees that the function can be recovered by integrating its derivative (via the Fundamental Theorem of Calculus in the Lebesgue sense). It also preserves sets of measure zero, which matters in measure theory.

Properties of continuous functions

These theorems are the payoff for establishing continuity. They give you powerful conclusions about function behavior.

Intermediate value theorem

If $f$ is continuous on $[a,b]$ and $y$ is any value between $f(a)$ and $f(b)$, then there exists some $c \in [a,b]$ with $f(c) = y$.

In plain terms: a continuous function can't skip over any values. If $f(0) = -3$ and $f(5) = 7$, then $f$ must hit every value between $-3$ and $7$ at least once. This is the standard tool for proving that equations have solutions (root-finding). If you can show a continuous function is negative at one point and positive at another, a root must exist between them.

Extreme value theorem

A continuous function on a closed, bounded interval $[a,b]$ attains both a maximum and a minimum value somewhere on that interval.

Both conditions matter. On an open interval like $(0,1)$, the function $f(x) = \frac{1}{x}$ is continuous but has no maximum. On an unbounded interval like $[0, \infty)$, the function $f(x) = x$ is continuous but has no maximum. You need the domain to be closed and bounded.

Mean value theorem

If $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists some $c \in (a,b)$ such that:

$f'(c) = \frac{f(b) - f(a)}{b - a}$

This says that somewhere between $a$ and $b$, the instantaneous rate of change equals the average rate of change over the whole interval. Note that this theorem requires differentiability (a stronger condition than continuity) on the open interval, plus continuity on the closed interval.

Discontinuities

When continuity fails at a point, the type of failure matters. Different discontinuities have different severity and different implications.

Removable discontinuity

The limit exists at the point, but either the function isn't defined there or its value doesn't match the limit. You can "fix" the function by redefining it at that single point.

Example: $f(x) = \frac{x^2 - 1}{x - 1}$ at $x = 1$. The function simplifies to $x + 1$ everywhere except $x = 1$, where it's undefined. The limit as $x \to 1$ is $2$, so defining $f(1) = 2$ removes the discontinuity.

Jump discontinuity

Both the left-hand and right-hand limits exist, but they aren't equal. The function "jumps" from one value to another. No single redefinition at the point can fix this.

Step functions and many piecewise-defined functions exhibit jump discontinuities. For instance, the floor function $\lfloor x \rfloor$ jumps at every integer.

Infinite discontinuity

The function approaches $\infty$ or $-\infty$ near the point. This typically shows up as a vertical asymptote.

Example: $f(x) = \frac{1}{x}$ as $x \to 0$. The function blows up in opposite directions from the left and right. No finite redefinition can repair this.

Continuity on intervals

Open vs closed intervals

• On an open interval $(a,b)$, continuity means the function is continuous at every interior point. The function doesn't need to be defined at $a$ or $b$.
• On a closed interval $[a,b]$, continuity additionally requires right-hand continuity at $a$ and left-hand continuity at $b$.

The distinction matters because the powerful theorems (extreme value theorem, etc.) require closed, bounded intervals.

One-sided continuity

• Left-hand continuity at $a$: $\lim_{x \to a^-} f(x) = f(a)$
• Right-hand continuity at $a$: $\lim_{x \to a^+} f(x) = f(a)$

A function is continuous at $a$ if and only if it's both left-continuous and right-continuous there. One-sided continuity is especially useful at endpoints of intervals and at transition points of piecewise functions.

Piecewise continuity

Piecewise functions are defined by different formulas on different parts of their domain. To check continuity:

1. Verify each piece is continuous on its own sub-interval.
2. At each transition point, check that the left-hand limit (from the piece on the left) equals the right-hand limit (from the piece on the right) and that both equal the function's value at that point.

If the pieces don't "match up" at a transition point, you get a jump or removable discontinuity there.

Continuity and limits

Relationship to limits

Continuity and limits are tightly linked, but they're not the same thing. The limit $\lim_{x \to a} f(x)$ describes what value $f(x)$ approaches, while $f(a)$ is the value the function actually takes. Continuity is the condition that these two agree.

• A limit can exist at a point where the function is discontinuous (removable discontinuity).
• If a function is continuous at $a$, then the limit at $a$ automatically exists and equals $f(a)$.

Left-hand vs right-hand limits

• The left-hand limit $\lim_{x \to a^-} f(x)$ considers only values of $x$ less than $a$.
• The right-hand limit $\lim_{x \to a^+} f(x)$ considers only values of $x$ greater than $a$.

The two-sided limit exists if and only if both one-sided limits exist and are equal. When they disagree, you have a jump discontinuity.

Continuity at infinity

This concerns the behavior of $f(x)$ as $x \to \infty$ or $x \to -\infty$. If $\lim_{x \to \infty} f(x) = L$ for some finite $L$, the function has a horizontal asymptote at $y = L$. Strictly speaking, there's no "point at infinity" in the real numbers, so this isn't continuity in the usual sense. It's better understood as a statement about the function's end behavior.

Continuity in multiple dimensions

When functions take inputs from $\mathbb{R}^n$ (multiple variables), continuity becomes more subtle.

Partial continuity

A function $f(x, y)$ is partially continuous if it's continuous in each variable separately while the other is held fixed. This is a weaker condition than you might expect: a function can be continuous in $x$ for every fixed $y$, and continuous in $y$ for every fixed $x$, yet still be discontinuous as a function of both variables together.

Joint continuity

Joint continuity means the function is continuous with respect to all variables at once. The function must approach the same limit no matter what path you take through the input space toward the point. This is the "real" notion of continuity for multivariable functions, and it implies partial continuity (but not vice versa).

Directional continuity

This examines continuity along specific directions or paths in the input space. You can think of it as a generalization of one-sided continuity to higher dimensions. A function might be continuous along every straight line through a point but still fail to be jointly continuous there (because some curved path reveals a discontinuity).

Applications of continuity

In calculus

Continuity is the foundation that makes calculus work. Differentiation requires continuity (every differentiable function is continuous). The Fundamental Theorem of Calculus requires continuity of the integrand. Techniques like L'Hôpital's rule, Taylor series, and optimization all depend on continuity assumptions.

In topology

Topology generalizes continuity beyond functions on $\mathbb{R}$. In a topological space, a function is continuous if the preimage of every open set is open. This abstract definition recovers the epsilon-delta version for real-valued functions but also applies to spaces with very different structures. Homeomorphisms (bijective functions where both the function and its inverse are continuous) are the central notion of equivalence in topology.

In real-world modeling

Physical quantities like temperature, pressure, and position typically change continuously. Modeling them with continuous functions lets you apply calculus to predict behavior, find optimal values, and analyze rates of change. Discontinuous models are sometimes needed (phase transitions, switching circuits), but continuity is the default assumption in physics, engineering, and economics because it reflects the smooth behavior of most natural systems.

Testing for continuity

Graphical methods

Look at the graph and check for:

• Holes (removable discontinuities)
• Jumps (the graph suddenly shifts to a different value)
• Vertical asymptotes (the graph shoots off toward infinity)

If you can trace the graph through a region without lifting your pencil, the function is continuous there. This method is quick but imprecise; it won't catch subtle issues.

Algebraic methods

To prove continuity at a point $a$ algebraically:

1. Confirm $f(a)$ is defined.
2. Compute $\lim_{x \to a} f(x)$ using limit laws, factoring, rationalization, or other techniques.
3. Verify that the limit equals $f(a)$.

For rational functions, check where the denominator is zero. For piecewise functions, focus on the transition points and compute one-sided limits.

Numerical methods

When algebraic methods are impractical, you can evaluate the function at inputs increasingly close to the point of interest (from both sides) and see whether the outputs converge to the function's value. This approach is approximate and can't constitute a proof, but it's useful for exploring unfamiliar functions or checking your work computationally.

Continuity of common functions

Polynomial functions

Polynomials are continuous on all of $\mathbb{R}$. This follows directly from the limit laws: limits of sums, products, and constant multiples of continuous functions are continuous. Since every polynomial is built from these operations applied to $x$ and constants, continuity is guaranteed everywhere.

Rational functions

A rational function $\frac{p(x)}{q(x)}$ (where $p$ and $q$ are polynomials) is continuous at every point where $q(x) \neq 0$. At points where $q(x) = 0$:

• If $p(x)$ and $q(x)$ share a common factor that vanishes there, you may have a removable discontinuity.
• Otherwise, you typically get a vertical asymptote (infinite discontinuity).

Trigonometric functions

$\sin(x)$ and $\cos(x)$ are continuous on all of $\mathbb{R}$. Functions like $\tan(x) = \frac{\sin(x)}{\cos(x)}$ are continuous wherever they're defined, but have infinite discontinuities where the denominator is zero (at odd multiples of $\frac{\pi}{2}$). The same pattern holds for $\cot(x)$, $\sec(x)$, and $\csc(x)$.

Advanced topics in continuity

Lipschitz continuity

A function is Lipschitz continuous if there exists a constant $K \geq 0$ such that:

$|f(x) - f(y)| \leq K|x - y|$

for all $x, y$ in the domain. This bounds how fast the function can change: the slope between any two points never exceeds $K$. Lipschitz continuity implies uniform continuity. It's a key condition in the Picard-Lindelöf theorem, which guarantees existence and uniqueness of solutions to ordinary differential equations.

Hölder continuity

Hölder continuity generalizes Lipschitz continuity by allowing a fractional exponent:

$|f(x) - f(y)| \leq K|x - y|^\alpha$

for some constant $K$ and exponent $0 < \alpha \leq 1$. When $\alpha = 1$, this reduces to Lipschitz continuity. Smaller values of $\alpha$ allow rougher functions. Hölder conditions appear in the study of partial differential equations and fractal geometry, where functions may be continuous but not smooth.

Continuity in metric spaces

In a general metric space, continuity is defined using the metric (distance function) in place of absolute values. A function $f: (X, d_X) \to (Y, d_Y)$ is continuous at $a$ if for every $\epsilon > 0$, there exists $\delta > 0$ such that $d_X(x, a) < \delta$ implies $d_Y(f(x), f(a)) < \epsilon$. This is the natural generalization of the epsilon-delta definition and provides the framework for studying continuity in complex analysis, functional analysis, and abstract topology.