Problem 18 - Basic Analysis: Introduction to Real Analysis

Question

Suppose \(f:[-1,1] \rightarrow \mathbb{R}\) is a function continuous at all \(x \in[-1,1] \backslash\\{0\\} .\) Show that for every \(\varepsilon\) such that \(0<\varepsilon<1,\) there exists a function \(g:[-1,1] \rightarrow \mathbb{R}\) continuous on all of [-1,1] , such that \(f(x)=g(x)\) for all \(x \in[-1,-\varepsilon] \cup[\varepsilon, 1]\), and \(|g(x)| \leq|f(x)|\) for all \(x \in[-1,1]\).

Step-by-Step Solution

Answer

Define a continuous function \(g(x)\) that matches \(f(x)\) outside \((-\varepsilon, \varepsilon)\) and smoothly transitions within, while maintaining \(|g(x)| \leq |f(x)|\).

Step 1: Understanding the Problem

We are given a function \(f\) which is continuous at all points in the interval \([-1, 1]\) except at \(x = 0\). We need to construct another function \(g\) which is continuous over the entire interval \([-1, 1]\) and equals \(f\) outside the open interval \((-\varepsilon, \varepsilon)\). Meanwhile, \(g\) should always have values less than or equal to the absolute values of \(f\).

Step 2: Define the Function g

Define the function \(g(x)\) as equal to \(f(x)\) on the intervals \([-1, -\varepsilon]\) and \([\varepsilon, 1]\). For \(x\) in the open interval \((-\varepsilon, \varepsilon)\), construct \(g(x)\) such that it smoothly transitions between \(f(-\varepsilon)\) and \(f(\varepsilon)\). A simple way to construct \(g\) in this interval is to use a polynomial or trigonometric function to interpolate between these boundary points.

Step 3: Ensure Continuity of g

To ensure \(g(x)\) is continuous on \([-1, 1]\), make sure the transition in the definition for \(g\) is smooth at \(-\varepsilon\) and \(\varepsilon\). This can be achieved using a continuous piecewise function or a smooth interpolating polynomial like a cubic spline that is constant and equal to \(f\) outside \((-\varepsilon, \varepsilon)\).

Step 4: Satisfy the Condition |g(x)| ≤ |f(x)|

Design the transition function such that \(|g(x)| \leq |f(x)|\) for all \(x \in [-1, 1]\), including the interval \((-\varepsilon, \varepsilon)\). Because \(g(x)\) interpolates linearly or smoothly between \(f(-\varepsilon)\) and \(f(\varepsilon)\), it remains bounded by \(|f(x)|\).

Step 5: Verify the Solution

Check the constructed function \(g\) to ensure it meets all conditions: it is continuous over the entire interval, matches \(f(x)\) in the required intervals, and satisfies \(|g(x)| \leq |f(x)|\) across the entire domain.

Key Concepts

Continuous Functions

Continuous functions are essential in real analysis and describe functions that do not have any jumps, breaks, or holes in their graphs. Such functions are smooth, and you can draw them without lifting your pen from the paper. A function \( f \) is considered continuous at a point \( x = c \) if:

• The limit of \( f(x) \) as \( x \) approaches \( c \) exists.
• \( f(c) \) is defined.
• The limit of \( f(x) \) as \( x \) approaches \( c \) is equal to \( f(c) \).

If a function meets these conditions at all points in its domain, it’s continuous on that domain. For instance, consider polynomial functions or sine and cosine functions, which are continuous everywhere on the real line.
In the original exercise, we find a function \( f \) that is continuous everywhere in the interval \([-1, 1]\), except possibly at \( x = 0 \). We thus need another function \( g \) that makes the overall function continuous over the entire interval \([-1, 1]\). This shows the practical importance of continuous functions in filling in gaps for non-continuous points, ensuring smoothness and consistency.

Piecewise Functions

Piecewise functions are constructed using different expressions for different parts of their domain. They allow different rules or formulas to define specific sections of the function’s graph. Imagine a situation where one formula is suitable for the left side of a graph and another for the right. Here, a piecewise function efficiently "stitches" these different expressions together to give a full picture of the entire domain.
In our context, the function \( g \) is built as a piecewise function:

• On the intervals \([-1, -\varepsilon] \) and \([\varepsilon, 1]\), \( g \) equals \( f \), meaning it takes the same values as \( f \).
• Within the gap \((-\varepsilon, \varepsilon)\), \( g \) must transition smoothly between \( f(-\varepsilon) \) and \( f(\varepsilon) \).

By defining \( g \) in this manner, we ensure it is continuous throughout \([-1, 1]\) without affecting the parts outside \((-\varepsilon, \varepsilon)\), demonstrating how piecewise functions can be exceptionally flexible in managing continuity and other properties.

Interpolation

Interpolation is the method of constructing new data points within a range of known data points. It is mainly used in mathematics and computer science to estimate values between two known values. In our exercise, interpolation helps create the continuous transition of the function \( g \) in the interval \((-\varepsilon, \varepsilon)\).
There are different types of interpolation methods, such as:

• Linear interpolation, which connects two data points with a straight line.
• Polynomial interpolation, using a polynomial equation to pass through multiple data points. Cubic splines, a form of polynomial interpolation, are often used because they ensure smoothness.

For the function \( g \), an interpolation technique ensures that \( g(x) \) not only bridges the values of \( f \) at \(-\varepsilon\) and \(\varepsilon\) smoothly, but also adheres to the condition \(|g(x)| \leq |f(x)|\). This approach maintains continuity in \([-1, 1] \), showing how interpolation is a powerful tool to achieve a smooth functional transition.