Question

In the following cases, show that \(f\) is uniformly continuous on \(B \subseteq E^{1}\), but only continuous (in the ordinary sense) on \(D,\) as indicated, with \(0

Short answer

In each case, the function is uniformly continuous on \(B\) but only continuous on \(D\).

Step-by-step solution

Understanding Uniform Continuity on Set B

For a function to be uniformly continuous on a set \(B\), given any \(\epsilon > 0\), there must exist a \(\delta > 0\) such that for all \(x, y \in B\), if \(|x - y| < \delta\) then \(|f(x) - f(y)| < \epsilon\). Let's examine each case under this criteria starting with the provided set \(B\).

Case (a) Uniform Continuity

For \(f(x) = \frac{1}{x^2}\) on \(B = [a, +\infty)\), since \(x^2\) is always positive and increasing, \(f(x)\) decreases as \(x\) increases, creating a smoother curve. For large \(x\), small changes in \(x\) result in even smaller changes in \(f(x)\), ensuring uniform continuity.

Case (a) Ordinary Continuity

On \(D = (0, 1)\), as \(x\) approaches 0, \(\frac{1}{x^2}\) tends towards infinity. This extreme increase creates a significant change in function value for a small change in \(x\), preventing uniform continuity but allowing ordinary continuity.

Case (b) Uniform Continuity

For \(f(x) = x^2\) on \(B = [a, b]\) which is a closed and bounded interval, it is uniformly continuous based on the Heine-Cantor theorem, as continuous functions on compact sets are uniformly continuous.

Case (b) Ordinary Continuity

On \(D = [a, +\infty)\), \(f(x)\) remains continuous as it just stretches and grows indefinitely, thus maintaining ordinary continuity.

Case (c) Uniform Continuity

For \(f(x) = \sin\frac{1}{x}\) on \(B = [a, +\infty)\), the function oscillates increasingly slowly, which stabilizes changes created by \(1/x\), ensuring uniform continuity.

Case (c) Ordinary Continuity

On \(D = (0, 1)\), as \(x\) approaches zero, \(1/x\) causes the oscillations of the sine function to increase rapidly, maintaining only ordinary continuity.

Case (d) Uniform Continuity

For \(f(x) = x \cos x\) on \(B = [a, b]\), a compact set, the function is uniformly continuous by the Heine-Cantor theorem.

Case (d) Ordinary Continuity

On \(D = [a, +\infty)\), as \(x\) grows, the amplitude \(x\) of the wave \(\cos x\) increases as well, thus making it lose uniform continuity but retaining ordinary continuity.

Key concepts

Ordinary Continuity

Continuity, in its ordinary sense, means that small changes in the input result in small changes in the output. This concept does not require the function to have a 'uniform' behavior across its domain. Ordinary continuity particularly emphasizes that for any \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that if \( |x - a| < \delta \), then \( |f(x) - f(a)| < \epsilon \). Here,

• The \( \delta \) is specific to \( a \), meaning its value depends on where in the domain you are.
• This definition allows for situations where particularly tricky spots in the domain might require especially small \( \delta \) values to keep the function close to continuous around a point.

This could be problematic in segments where the function rapidly changes, such as near asymptotes or within oscillating functions, like in \( f(x) = \frac{1}{x^2} \) as \( x \) approaches 0.
Such ordinary continuity doesn't mean the function will behave uniformly across the entire domain, which is where uniform continuity comes in.

Heine-Cantor Theorem

The Heine-Cantor theorem is quite powerful when dealing with uniformly continuous functions over closed and bounded sets. Simply put, the theorem states that if a function is continuous on a compact space, which means closed and bounded in \( E^1 \), then it is uniformly continuous on that space.
To break it down:

• "Closed" means that the set includes its boundary or limit points.
• "Bounded" means the set doesn't stretch to infinity in any direction.

So when a function like \( f(x) = x^2 \) is defined on a compact interval \([a, b]\), it is both neatly contained, and thus its behavior uniform.
Functions such as \( f(x) = x \cos x \) on compact domains utilize this theorem, ensuring that their behavior is predictable and uniform, justifying their uniform continuity by this theorem. It fundamentalizes the importance of limiting where discontinuities creep in, maintaining smooth behavior throughout.

Continuous Functions

Continuous functions are those that smoothly transition without any sharp breaks or jumps—any change in the input results in a proportional change in the output. Continuous functions, by definition, adhere to the ordinary continuity rules over their entire domain.
Factors that contribute to a function being continuous might include:

• The lack of vertical asymptotes or holes in its graph.
• Unbroken graphs without sharp corners.
• Predictable behavior expected from well-behaved polynomial, trigonometric, and exponential functions within their defined intervals.

But even among continuous functions, not every function is uniformly continuous. For example, \( f(x) = x^2 \) is continuous everywhere but only uniformly continuous over certain intervals, notably those that are bounded and closed, as stipulated by the Heine-Cantor theorem.
This distinction matters especially when functions "blow up" or oscillate rapidly, leading them to only locally meet the standard requirements without global uniformity.

Sine Function Oscillation

The sine function, especially in forms such as \( \sin \frac{1}{x} \), is an intriguing specimen for studying oscillatory behavior in continuous functions. Oscillation refers to the repeated upward and downward waving of the function's graph, which becomes rapid as \(x\) approaches 0 in \( \sin \frac{1}{x} \).
In educational terms, here's why it matters:

• As \(x\) gets closer to 0, \( \frac{1}{x} \) becomes large, making the sine function oscillate faster, increasing the frequency of the waves.
• This behavior makes \( \sin \frac{1}{x} \) challenging to predict uniformly, leading to ordinary continuity but losing uniform continuity near 0.
• The value of \( \delta \) in the continuity definition has to adapt to manage the increased oscillation.
• While on broader intervals like \([a, +\infty)\), the waves stretch out, simplifying the function's behavior and making uniform continuity possible.

Thus, understanding sine function oscillation helps demystify the shift from stable to erratic behaviors in trigonometric functions and their continuity.