Uniform Continuity
Welcome, students 👋 This lesson focuses on one of the most important ideas in Real Analysis: uniform continuity. At first, it looks like ordinary continuity with a tiny twist, but that twist changes everything. By the end of this lesson, you should be able to explain what uniform continuity means, tell it apart from ordinary continuity, and use it in proofs and examples from the Midterm 1 unit.
What you will learn
In this lesson, students, you will learn to:
- explain the meaning of uniform continuity in clear language,
- use the formal definition with $ epsilon$ and $ delta$ ideas correctly,
- compare uniform continuity with ordinary continuity,
- prove that common functions are uniformly continuous on certain sets,
- recognize why boundedness and interval shape matter, and
- connect uniform continuity to the broader themes of Midterm 1, including continuity and connectedness.
Think of this topic as learning how a function behaves when you zoom in anywhere on its domain 🔍. Ordinary continuity asks whether the graph behaves nicely at one point. Uniform continuity asks whether the same kind of control works everywhere at once.
The idea behind uniform continuity
For ordinary continuity at a point $a$, the idea is this: if $x$ is close enough to $a$, then $f(x)$ is close to $f(a)$. The exact size of how close depends on the point $a$.
Uniform continuity removes that dependence on the point. A function $f$ is uniformly continuous on a set $E$ if for every $ epsilon > 0$, there exists a $ delta > 0$ such that for all $x,y in E$,
$$
|x-y| < delta \implies |f(x)-f(y)| < epsilon.
$$
The key word is uniform. The same $ delta$ works for every pair of points in the domain. students, this means the function cannot suddenly become much steeper in some region and still be uniformly continuous there.
A good way to picture this is with a thermostat 🌡️. Ordinary continuity says, “At each room in the house, I can adjust the temperature control separately.” Uniform continuity says, “One setting works for the entire house.” That is a stronger and more global condition.
Formal definition and how to read it
The definition uses the same logic as continuity, but the quantifiers matter a lot. Written carefully, $f:E to \mathbb{R}$ is uniformly continuous if
$$
$\forall$ epsilon > 0\, $\exists$ delta > 0\, $\forall$ x,y in E,\ |x-y| < delta \Rightarrow |f(x)-f(y)| < epsilon.
$$
Compare this with continuity at a point $a$:
$$
$\forall$ epsilon > 0\, $\exists$ delta > 0\, $\forall$ x in E,\ |x-a| < delta \Rightarrow |f(x)-f(a)| < epsilon.
$$
The difference is subtle but important. In uniform continuity, the same $ delta$ must work for every point in the domain. In pointwise continuity, the choice of $ delta$ may depend on the point $a$.
When proving uniform continuity, students, a common strategy is to start from
$$
|f(x)-f(y)|
$$
and try to bound it in terms of
$$
|x-y|.
$$
If you can show something like
$$
|f(x)-f(y)| leq M|x-y|
$$
for some constant $M$, then $f$ is uniformly continuous. This is called a Lipschitz condition, and it is stronger than uniform continuity.
Examples that are uniformly continuous
Let’s look at some familiar functions and domains.
Example 1: Linear functions
A linear function such as
$$
$ f(x)=mx+b$
$$
is uniformly continuous on all of $\mathbb{R}$. Indeed,
$$
$|f(x)-f(y)|=|m||x-y|.$
$$
So choosing $ delta = epsilon/|m|$ when $m neq 0$ works for all $x,y$. If $m=0$, the function is constant, so it is uniformly continuous even more easily.
Example 2: The square root function on $[0,\infty)$
The function
$$
$ f(x)=\sqrt{x}$
$$
is uniformly continuous on $[0,\infty)$. A useful estimate is
$$
$|\sqrt{x}-\sqrt{y}|=\frac{|x-y|}{\sqrt{x}+\sqrt{y}}.$
$$
This denominator can get small near $0$, so the function is not Lipschitz on all of $[0,\infty)$, but it is still uniformly continuous. One way to see this is by using the fact that for $x,y ge 0$,
$$
$|\sqrt{x}-\sqrt{y}|^2 leq |x-y|.$
$$
Then if $|x-y|<\u0000delta$, we get
$$
$|\sqrt{x}-\sqrt{y}|<\sqrt{\u0000delta}.$
$$
So choosing $ delta = epsilon^2$ works.
Example 3: Continuous functions on closed and bounded intervals
A major theorem in Real Analysis says that every continuous function on a closed and bounded interval $[a,b]$ is uniformly continuous. This is often called the Heine-Cantor theorem.
So if $f$ is continuous on $[0,1]$, then it is automatically uniformly continuous there. This theorem is extremely important because it lets you move from local continuity to global control on compact sets.
For example, the function
$$
$ f(x)=x^2$
$$
is uniformly continuous on $[0,1]$ because it is continuous on a closed and bounded interval.
Examples that are not uniformly continuous
Now let’s see where uniform continuity fails. These examples are just as important as the positive ones.
Example 1: $f(x)=x^2$ on all of $\mathbb{R}$
The function $f(x)=x^2$ is continuous everywhere, but it is not uniformly continuous on $\mathbb{R}$. Why? Because the graph gets steeper and steeper as $|x|$ grows.
To show this, choose pairs of points that are very close together but far out on the number line. Let
$$
$ x_n=n, \qquad y_n=n+\frac{1}{n}.$
$$
Then
$$
$|x_n-y_n|=\frac{1}{n} \to 0,$
$$
but
$$
|f(x_n)-f(y_n)|=$\left|$n^2-$\left($n+$\frac{1}{n}$$\right)^2$$\right|$=$\left|2$+$\frac{1}{n^2}$$\right|$,
$$
which does not go to $0$. So there is no single $ delta$ that works everywhere.
Example 2: $f(x)=\frac{1}{x}$ on $(0,1)$
The function
$$
$ f(x)=\frac{1}{x}$
$$
is continuous on $(0,1)$ but not uniformly continuous there. The problem is behavior near $0$, where the function blows up.
Take
$$
$ x_n=\frac{1}{n}, \qquad y_n=\frac{1}{n+1}.$
$$
Then
$$
$|x_n-y_n|=\frac{1}{n(n+1)} \to 0,$
$$
but
$$
$\left|\frac{1}{x_n}-\frac{1}{y_n}\right|=|n-(n+1)|=1.$
$$
So the outputs stay separated by $1$ even though the inputs get arbitrarily close.
Why the interval matters
students, one of the biggest lessons in this topic is that the shape of the domain matters a lot. A function can be continuous everywhere on a set and still fail to be uniformly continuous if the set is not well behaved.
The classic theorem says:
- continuous on $[a,b]$ $ Rightarrow$ uniformly continuous,
- continuous on an open or unbounded interval does not automatically imply uniformly continuous.
For instance, $f(x)=x^2$ is uniformly continuous on $[0,1]$ but not on $\mathbb{R}$. The function did not change; only the domain changed. That is a big Real Analysis idea: properties of functions depend not just on the formula, but also on the set where the function lives.
This is one reason connectedness and compactness appear in the same unit. Closed intervals in $\mathbb{R}$ are connected and compact, and these structural properties help guarantee nice behavior like uniform continuity.
How to prove uniform continuity in practice
When you are asked to prove a function is uniformly continuous, students, there are a few common tools:
- Direct $ epsilon$-$\u0000delta$ proof
- Start with $|f(x)-f(y)|$.
- Try to make it less than $ epsilon$ whenever $|x-y|<\u0000delta$.
- Use a known theorem
- If the function is continuous on $[a,b]$, apply Heine-Cantor.
- Use algebraic inequalities
- For polynomials or roots, rewrite and estimate carefully.
- Use a contradiction or sequence argument
- If the function is not uniformly continuous, you can often find sequences $x_n,y_n$ with $|x_n-y_n|\to 0$ but $|f(x_n)-f(y_n)|$ not tending to $0$.
For example, suppose $f(x)=\sin x$ on $\mathbb{R}$. Since
$$
$|\sin x-\sin y|\leq |x-y|,$
$$
it is uniformly continuous on all of $\mathbb{R}$. This is a strong and easy-to-use estimate.
Conclusion
Uniform continuity is a stronger version of continuity that asks for one common control value $ delta$ to work everywhere in the domain. This makes it a global property, not just a point-by-point one. In this lesson, students, you saw the formal definition, compared it with ordinary continuity, and explored examples that do and do not satisfy it.
Uniform continuity is important in Midterm 1 because it connects the local idea of continuity to global behavior on intervals. It also works naturally with the big Real Analysis themes of connectedness, compactness, and rigorous proof. When you understand uniform continuity, you are learning how to control a function consistently across an entire set, which is a powerful idea in advanced mathematics.
Study Notes
- Uniform continuity means one $ delta$ works for all points in the domain.
- Ordinary continuity at a point allows $ delta$ to depend on that point.
- Formal definition:
$$
$\forall$ epsilon > 0\, $\exists$ delta > 0\, $\forall$ x,y in E,\ |x-y|<\u0000delta \Rightarrow |f(x)-f(y)|< epsilon.
$$
- Every Lipschitz function is uniformly continuous.
- Every continuous function on a closed and bounded interval $[a,b]$ is uniformly continuous.
- A continuous function on an open or unbounded set may fail to be uniformly continuous.
- Common non-examples: $f(x)=x^2$ on $\mathbb{R}$ and $f(x)=1/x$ on $(0,1)$.
- Common examples: linear functions on $\mathbb{R}$, $f(x)=\sin x$ on $\mathbb{R}$, and continuous functions on $[a,b]$.
- To disprove uniform continuity, look for sequences $x_n,y_n$ with $|x_n-y_n|\to 0$ but $|f(x_n)-f(y_n)|\not\to 0$.
- Uniform continuity is a key bridge between pointwise continuity and global behavior in Real Analysis.