Question

The functions \(f_{n}: A \rightarrow\left(T, \rho^{\prime}\right), A \subseteq(S, \rho)\) are said to be equicontinuous at \(p \in A\) iff $$ (\forall \varepsilon>0)(\exists \delta>0)(\forall n)\left(\forall x \in A \cap G_{p}(\delta)\right) \quad \rho^{\prime}\left(f_{n}(x), f_{n}(p)\right)<\varepsilon $$ Prove that if so, and if \(f_{n} \rightarrow f\) (pointwise) on \(A\), then \(f\) is continuous at \(p\).

Short answer

If \( f_n \) is equicontinuous and \( f_n \rightarrow f \), then \( f \) is continuous at \( p \).

Step-by-step solution

Understand Definitions

Equicontinuity at a point \( p \) means that for every \( \varepsilon > 0 \), there exists a \( \delta > 0 \) such that for any index \( n \) and any \( x \) in the \( \delta \)-neighborhood of \( p \), \( \rho' (f_n(x), f_n(p)) < \varepsilon \). Pointwise convergence \( f_n \rightarrow f \) means that for each \( x \in A \), \( \lim_{n \to \infty} f_n(x) = f(x) \). Continuity of \( f \) at \( p \) means that for every \( \varepsilon > 0 \), there exists \( \delta > 0 \) such that whenever \( x \in A \) and \( \rho(x, p) < \delta \), it holds that \( \rho'( f(x), f(p) ) < \varepsilon \).

Choose \( \varepsilon > 0 \)

Given that we need \( f \) to be continuous at \( p \), we begin by choosing an arbitrary \( \varepsilon > 0 \). According to our goal, we need to find a suitable \( \delta > 0 \) based on this epsilon.

Use Equicontinuity

Because the sequence \( \{f_n\} \) is equicontinuous at \( p \), for the \( \varepsilon/3 \) above, there exists a \( \delta_1 > 0 \) such that for all \( n \) and all \( x \in A \cap G_p(\delta_1) \), \( \rho'(f_n(x), f_n(p)) < \varepsilon/3 \).

Use Pointwise Convergence

Since \( f_n \rightarrow f \) pointwise, for \( x = p \), there exists an \( N_0 \) such that for all \( n \geq N_0 \), \( \rho'(f_n(p), f(p)) < \varepsilon/3 \). Similarly, for a fixed \( x \) (later within \( \delta \) of \( p \)), there exists an \( N_1 \) such that for all \( n \geq N_1 \), \( \rho'(f_n(x), f(x)) < \varepsilon/3 \).

Combine Conditions

Take \( N = \max(N_0, N_1) \) and let \( \delta = \delta_1 \). For this \( \delta \), whenever \( n \geq N \), and \( x \in A \cap G_p(\delta) \), we have \( \rho'(f(x), f(p)) < \varepsilon/3 + \varepsilon/3 + \varepsilon/3 = \varepsilon \) by triangle inequality applied as follows: \( \rho'(f(x), f(p)) \leq \rho'(f(x), f_n(x)) + \rho'(f_n(x), f_n(p)) + \rho'(f_n(p), f(p)) \).

Key concepts

Pointwise Convergence

Pointwise convergence is a fundamental concept in mathematical analysis where we focus on the behavior of a sequence of functions at each individual point in their domain. Let's break it down simply: if we have a sequence of functions \( f_n \) defined on the same set \( A \), this sequence is said to pointwise converge to a function \( f \) if, for every point \( x \) in \( A \), the values of \( f_n(x) \) get closer and closer to \( f(x) \) as \( n \) goes to infinity.
To see this in action, imagine standing at each point \( x \) in \( A \) as \( n \) increases. If at every point \( x \), \( \lim_{n \to \infty} f_n(x) = f(x) \), then \( f_n \) pointwise converges to \( f \).

• This convergence is verified individually one point at a time.
• It doesn't require any uniform approach across the whole domain.

This is different from uniform convergence, which demands the functions converge in a similar manner across the entire domain. In many problems, pointwise convergence helps us identify the limiting function \( f \) from a sequence of approximations.

Continuity

Continuity is a key property of functions that describes smooth, uninterrupted behavior within its domain. A function \( f \) is continuous at a point \( p \) if, intuitively, when you are close to \( p \), the function's value \( f(x) \) is close to \( f(p) \).
For a formal understanding, let's dissect the definition: a function \( f \) is continuous at \( p \) if for every small latitude \( \varepsilon > 0 \), there's a bandwidth \( \delta > 0 \) such that whenever \( x \) is within this \( \delta \)-range of \( p \), the difference \( \rho'(f(x), f(p)) \) is within \( \varepsilon \).
In simple terms:

• Choose any tiny closeness \( \varepsilon \) on the output values.
• Find a corresponding closeness \( \delta \) on input values.
• Ensure that moving around \( p \) in that range doesn't disrupt the closeness \( \varepsilon \).

This provides the assurance that the function does not "jump" or "break" as you approach \( p \). Understanding continuity is fundamental in calculus and advanced analysis as it ensures that functions behave well under limits, derivations, and integrations.

Neighborhood

The concept of a neighborhood is central in understanding how analysis approaches behavior of functions around points. In mathematics, a neighborhood of a point \( p \) in a space \( A \) is essentially a "bubble" or "area" around \( p \).
More precisely, an \( \delta \)-neighborhood of a point \( p \) includes all points \( x \) in \( the \) space such that they are within \( \delta \) units of \( p \). Let's formalize this: for point \( p \), the neighborhood encompasses points \( x \) where \( \rho(x, p) < \delta \).
Consider these applications:

• Neighborhoods help define continuity: we can verify a function's continuity at \( p \) by checking its behavior inside its neighborhood.
• They provide little pockets of analysis to assess functions' behavior around their domain.
• Often described as the local environment of a point, helping to localize properties like limit and continuity.

In our context of equicontinuity, the neighborhood is crucial because it determines the set across which function values are compared, revealing uniform behavior in function sequences `
The mastery of neighborhoods enriches your understanding of function behavior, critical for a deeper dive into analysis.