Continuity

Continuity is about functions that do not have “jumps,” “holes,” or “sudden breaks.” In the earlier chapter on formal limits, you learned how to express the idea that $f(x)$ approaches a value as $x$ approaches a point. Continuity uses that language to describe the behavior of a function at, and around, a point.

In this chapter we:

• Use limits to define what it means for a function to be continuous.
• Classify common types of discontinuities.
• Look at continuity on intervals, not just at single points.
• State and use some basic theorems that rely on continuity.

We will not re-develop the formal limit definitions; we simply use them.

Continuity at a Point

Let $f$ be a function and let $a$ be a real number in its domain.

The function $f$ is continuous at $x = a$ if all three of the following conditions hold:

1. $f(a)$ is defined (so $a$ is in the domain of $f$),
2. $\displaystyle \lim_{x \to a} f(x)$ exists (as a finite real number),
3. $\displaystyle \lim_{x \to a} f(x) = f(a)$.

Symbolically:
$$
f \text{ is continuous at } a
\quad \Longleftrightarrow \quad
\lim_{x \to a} f(x) = f(a).
$$

You can think of this as saying:

• The limiting “prediction” from nearby $x$–values matches the actual function value at $a$.

If any one of the three conditions fails, $f$ is not continuous at $a$; we then say $f$ has a discontinuity at $a$.

One-Sided Continuity

Sometimes we only care about approaching from one side.

• $f$ is continuous from the right at $a$ if $f(a)$ is defined and
  $$
  \lim_{x \to a^+} f(x) = f(a).
  $$
• $f$ is continuous from the left at $a$ if $f(a)$ is defined and
  $$
  \lim_{x \to a^-} f(x) = f(a).
  $$

For ordinary continuity at an interior point $a$ of the domain, both one-sided limits must exist and be equal to $f(a)$.

On a closed interval like $[a,b]$, it is common to require:

• Continuity from the right at $a$,
• Continuity from the left at $b$,
• And usual (two-sided) continuity at interior points.

Types of Discontinuities

When a function is not continuous at a point, we often classify the “kind” of failure. The language here is descriptive; these labels help you understand what is going wrong and how to fix it (if possible).

Suppose $f$ is not continuous at $x = a$.

Removable Discontinuity (“Hole”)

$f$ has a removable discontinuity at $a$ if:

• $\displaystyle \lim_{x \to a} f(x)$ exists (finite),
• but either
• $f(a)$ is not defined, or
• $f(a)$ is defined but $f(a) \neq \displaystyle \lim_{x \to a} f(x)$.

In other words, the limit “wants” a certain value $L$ at $a$, but the function either misses that point or takes a different value.

Key idea: You could fix continuity at $a$ by redefining $f(a)$ to equal the limit $L$.

Example pattern (not fully worked out, just a type):

• A rational function where a factor cancels:
  $$
  f(x) = \frac{x^2 - 1}{x - 1}
  $$
  is undefined at $x = 1$ but behaves like $x+1$ elsewhere. The “hole” at $x=1$ is removable.

Jump Discontinuity

$f$ has a jump discontinuity at $a$ if:

• Both one-sided limits exist as finite numbers,
• But they are not equal:
  $$
  \lim_{x \to a^-} f(x) \neq \lim_{x \to a^+} f(x).
  $$

Geometrically, the graph “jumps” from one value to another as you pass $x = a$.

Key idea: No matter how you define $f(a)$, you cannot make the function continuous at $a$, because the left and right behavior disagree.

Jump discontinuities are typical in piecewise-defined functions.

Infinite Discontinuity

$f$ has an infinite discontinuity at $a$ if the function values grow without bound near $a$:

• At least one of the one-sided limits is $\infty$ or $-\infty$:
  $$
  \lim_{x \to a^-} f(x) = \pm \infty
  \quad \text{or} \quad
  \lim_{x \to a^+} f(x) = \pm \infty.
  $$

In this situation the graph shoots up or down vertically near $a$; $a$ is typically a vertical asymptote.

Again, no choice of $f(a)$ can fix this.

Oscillatory Discontinuity

Less common in basic examples, but important conceptually:

$f$ has an oscillatory discontinuity at $a$ if the function oscillates increasingly rapidly near $a$ and does not settle to any single limiting value.

In symbols, the limit $\displaystyle \lim_{x \to a} f(x)$ does not exist, not because it goes to $\pm \infty$, and not because of a jump, but because the function keeps changing value without approaching anything.

Key idea: Even arbitrarily close to $a$, $f(x)$ keeps swinging through different values.

Continuity on Intervals

So far, continuity was defined at a single point. We say a function is continuous on a set (typically an interval) if it is continuous at every point of that set.

• $f$ is continuous on an open interval $(a,b)$ if $f$ is continuous at every point $x$ with $a < x < b$.
• $f$ is continuous on a closed interval $[a,b]$ if:
• $f$ is continuous on $(a,b)$,
• $f$ is continuous from the right at $a$,
• $f$ is continuous from the left at $b$.

We often speak of “continuous functions” meaning functions that are continuous at every point in their entire domain.

In practical work, you frequently:

• Determine the domain of a formula (where it makes sense),
• Use known facts (below) to assert continuity on that domain,
• Then pay special attention to points where the formula might fail (e.g. denominator $=0$, square root of a negative number, piecewise boundaries, etc.).

Basic Continuity Properties

Many functions encountered in elementary calculus are continuous on large portions of $\mathbb{R}$. A few general facts are very useful:

• Polynomials are continuous everywhere on $\mathbb{R}$.
• Rational functions (quotients of polynomials) are continuous wherever the denominator is nonzero.
• Standard functions like $\sin x$, $\cos x$, $e^x$, $\ln x$ (on their natural domains) are continuous on those domains.

More generally, continuity is preserved under common algebraic operations:

Let $f$ and $g$ be continuous at $a$, and let $c$ be a constant. Then each of the following is also continuous at $a$:

• $(f + g)(x) = f(x) + g(x)$,
• $(f - g)(x) = f(x) - g(x)$,
• $(cf)(x) = c \cdot f(x)$,
• $(fg)(x) = f(x)g(x)$,
• If $g(a) \neq 0$, then
  $$
  \left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}
  $$
  is continuous at $a$.

These statements follow directly from limit laws: if $\lim_{x \to a} f(x)$ and $\lim_{x \to a} g(x)$ exist and equal $f(a)$ and $g(a)$, then the limits of their sums, products, etc., behave as expected and match the corresponding function values.

Composition and Continuity

Composition is especially important:

If $g$ is continuous at $a$ and $f$ is continuous at $g(a)$, then the composition
$$
h(x) = f(g(x))
$$
is continuous at $a$.

This is one of the main ways more complicated continuous functions are built from simpler ones.

Using Continuity: Evaluating Limits

One of the main advantages of continuity is that if $f$ is continuous at $a$, then:
$$
\lim_{x \to a} f(x) = f(a)
$$
by definition.

This allows you to evaluate many limits simply by direct substitution:

• Check that $a$ is in the domain of $f$ and that $f$ is continuous there (often by knowing the function type and its domain).
• If so, plug in $x = a$ directly in the expression.

If $f$ is not continuous at $a$, direct substitution typically does not give the limit (or does not even make sense). In those cases, you rely more heavily on limit techniques from the limits chapter and, later, derivative tools.

Continuity Theorems (Without Proof)

There are two central theorems about continuous functions on intervals that are used repeatedly in calculus. Here we state them informally and illustrate their meaning; their proofs belong in a more advanced treatment.

Assume $f$ is continuous on a closed interval $[a,b]$.

Intermediate Value Theorem (IVT)

If $f(a)$ and $f(b)$ are different values, then $f$ takes every intermediate value between $f(a)$ and $f(b)$ somewhere in the interval.

Formally: If $N$ is any number between $f(a)$ and $f(b)$, then there exists some $c$ in $(a,b)$ such that
$$
f(c) = N.
$$

Interpretation:

• A continuous graph cannot “skip” a height. To go from $f(a)$ to $f(b)$, it must pass through all intermediate heights.

Common uses:

• Showing that an equation $f(x) = 0$ has at least one solution in $(a,b)$ by checking $f(a)$ and $f(b)$ have opposite signs.
• Arguing that certain numbers (like roots of polynomials that cannot be expressed simply) exist even if we cannot write them exactly.

Extreme Value Theorem (EVT)

$f$ attains both a maximum and a minimum value on $[a,b]$.

Formally: There exist points $c$ and $d$ in $[a,b]$ such that
$$
f(c) \le f(x) \le f(d) \quad \text{for all } x \in [a,b].
$$

Interpretation:

• A continuous graph on a closed, bounded interval has a highest point and a lowest point (perhaps at the ends of the interval, perhaps in the interior).

This theorem is a foundational fact for optimization problems in calculus, where we want to know that best or worst values actually occur.

Summary

• Continuity at a point $a$ means the limit of $f(x)$ as $x \to a$ exists and equals $f(a)$.
• Discontinuities can be removable (holes), jumps, infinite (vertical asymptotes), or oscillatory.
• A function is continuous on an interval if it is continuous at every point there (with appropriate one-sided continuity at endpoints).
• Sums, products, quotients (where defined), and compositions of continuous functions are continuous.
• For continuous functions, many limits are evaluated simply by substitution.
• On closed intervals, continuous functions have powerful properties:
• IVT: they take every intermediate value between endpoint values.
• EVT: they achieve both maximum and minimum values.

These ideas about continuity form the foundation for defining derivatives and integrals and for justifying many of the methods used throughout calculus.