Question

Prove that if $f$ is uniformly continuous on $B,$ it is so on each subset $A\subseteq B$.

Short answer

If $f$ is uniformly continuous on $B$, it is so on every subset $A\subseteq B$.

Step-by-step solution

Understanding Uniform Continuity

Uniform continuity of a function means that for every $\epsilon >0$, there exists a $\delta >0$ such that for all $x,y\in B$, if $|x-y|<\delta$ then $|f(x)-f(y)|<\epsilon$. We need to show this condition holds for any subset $A$ of $B$.

Recognizing Subset Implications

Since $A\subseteq B$, it implies any $x,y\in A$ are also elements of $B$. Thus, if the condition for uniform continuity holds for all elements in $B$, it automatically holds for elements in $A$.

Defining the Problem for a Subset

On subset $A$, we need to ensure for every $\epsilon >0$, the same $\delta >0$ determined on $B$ ensures $|f(x)-f(y)|<\epsilon$ whenever $|x-y|<\delta$ for all $x,y\in A$.

Using Uniform Continuity on the Superset

Since for $B$, $\delta$ exists for every $\epsilon$, and $A$ is a subset of $B$, the same $\delta$ applies. Hence, any $x,y\in A$ satisfying $|x-y|<\delta$ will lead to $|f(x)-f(y)|<\epsilon$ just like in $B$.

Conclusion with Final Justification

Therefore, the condition for uniform continuity on $B$ applies directly to every subset $A$. Thus, $f$ is uniformly continuous on $A$ for all $A\subseteq B$.

Key concepts

Understanding Subsets

A subset is essentially a smaller collection of elements that reside within a larger set. Imagine having a set of numbers, say set $B$. If there is another set $A$ where every element in set $A$ also belongs to set $B$, then $A$ is called a subset of $B$. We denote this relationship with the symbol $A\subseteq B$.

It's crucial to note that every subset $A$ has elements that are also in set $B$, but set $B$ may contain additional elements not found in $A$. In simpler terms:

• All elements of $A$ are in $B$.
• Some elements of $B$ may not be in $A$.

Understanding subsets is fundamental in various mathematical concepts, especially in continuity and functions, as it allows us to apply properties established in larger sets to their respective subsets.

Exploring Functions

A function is a rule that assigns each element in one set to exactly one element in another set. For instance, a function $f$ might map numbers in a set $B$ to values, perhaps real numbers.

Functions are central to mathematics because they allow us to describe relationships between variables or sets. For example:

• Consider the function $f(x)=x^2$, mapping each number $x$ to its square.
• This can apply to a set of numbers, such as positive integers or real numbers, as the function domain.

Functions can vary widely, from simple equations like the one above to more complex interactions. Their core concept remains consistent, making them essential for proving various mathematical properties like continuity.

Crafting a Mathematical Proof

Mathematical proofs are rigorous arguments that are used to establish the truth of a mathematical statement. To construct a solid proof, you begin with assumptions or known facts and apply logical reasoning to arrive at the conclusion you want to demonstrate.

There are several approaches to crafting a proof, such as:

• Direct proof: Begins with known truths and logically progresses to the desired conclusion.
• Indirect proof (or proof by contradiction): Assumes that the statement to be proven is false, then shows that this assumption leads to a contradiction.

For proving uniform continuity, we typically employ direct proofs. Starting with the definition of uniform continuity, we deduce how the property holds in every subset $A\subseteq B$. Thus, mathematical proofs are vital for confirming that properties like uniform continuity in a superset $B$ transfer seamlessly to its subset $A$.

The Epsilon-Delta Definition of Uniform Continuity

When discussing continuity, the epsilon-delta definition provides a formal way to capture the idea of a function being smooth and consistent across its domain. Uniform continuity strengthens this concept by ensuring the consistency occurs across the entire set with ease.

• Epsilon ($\epsilon$): Represents a small, positive number indicating how closely we want the outputs $f(x)$ and $f(y)$ of the function to be.
• Delta ($\delta$): Corresponds to the tolerance in the inputs $|x-y|$, determining when the outputs' closeness is within our epsilon bounds.

In the realm of uniform continuity, once a suitable $\delta$ is found for a particular $\epsilon$ within a set $B$, the same $\delta$ can be applied to any subset $A$ of $B$ to achieve the desired closeness of function values. This property makes uniform continuity highly useful for handling functions over various domains!