Question

Let \(f:(A, \rho) \rightarrow(B, \sigma)\) be continuous on a subset \(U\) of \(A\). Let \(\bar{u}\) be in \(U\) and define the real-valued function \(g:(A, \rho) \rightarrow \mathbb{R}\) by $$ g(u)=\sigma(f(u), f(\bar{u})), \quad u \in U $$ (a) Show that \(g\) is continuous on \(U\). (b) Show that if \(U\) is compact, then \(g\) is uniformly continuous on \(U\). (c) Show that if \(U\) is compact, then there is a \(\widehat{u} \in U\) such that \(g(u) \leq g(\hat{u})\), \(u \in U\)

Short answer

Additionally, is there a point \(\hat{u}\) in U such that \(g(u) \leq g(\hat{u})\) for all \(u\) in U? Answer: Yes, g is continuous on U and uniformly continuous on U if U is compact. Moreover, there exists a point \(\hat{u}\) in U such that \(g(u) \leq g(\hat{u})\) for all \(u\) in U.

Step-by-step solution

Define continuity and uniformly continuity

To show that a function is continuous, we must show that for every point u in U, the limit of the function g as u approaches a point exists and is equal to the value of the function at that point. In other words, given any \(\epsilon > 0\), there exists a \(\delta > 0\) such that for all \(u\) in U with \(\rho(u, \overline{u}) < \delta\), we have \(|g(u) - g(\overline{u})| < \epsilon\). To show that a function is uniformly continuous, we need to know there is a single \(\delta > 0\) that works for all \(\overline{u}\) in U. In other words, given any \(\epsilon > 0\), there exists a \(\delta > 0\) such that for all \(u, \overline{u}\) in U with \(\rho(u, \overline{u}) < \delta\), we have \(|g(u) - g(\overline{u})| < \epsilon\).

Show that g is continuous on U

We know that f is continuous on U, which means for any \(\epsilon' > 0\), there exists a \(\delta' > 0\) such that for all \(u\) in U with \(\rho(u,\overline{u}) < \delta'\), we have \(\sigma(f(u),f(\overline{u})) < \epsilon'\). Now, let's choose \(\epsilon' = \epsilon\). Therefore, we have that \(|g(u) - g(\overline{u})| = |\sigma(f(u), f(\overline{u})) - \sigma(f(\overline{u}), f(\overline{u}))| = \sigma(f(u), f(\overline{u})) < \epsilon\). So, we can conclude that g is continuous on U.

Show that g is uniformly continuous on U if U is compact

If U is compact, we know that f is uniformly continuous on U. This means that for any \(\epsilon' > 0\), there exists a \(\delta' > 0\) such that for all \(u, \overline{u}\) in U with \(\rho(u, \overline{u}) < \delta'\), we have \(\sigma(f(u), f(\overline{u})) < \epsilon'\). Now, let's choose \(\epsilon' = \epsilon\). Therefore, we have that \(|g(u) - g(\overline{u})| = \sigma(f(u), f(\overline{u})) < \epsilon\). So, we can conclude that g is uniformly continuous on U if U is compact.

Show that there exists a \(\hat{u}\) in U such that \(g(u) \leq g(\hat{u})\) if U is compact

Since U is compact and g is continuous on U, g attains its maximum and minimum values on U. Let \(\hat{u}\) be a point in U where g attains its maximum value. Then, for all \(u\) in U, we have \(g(u) \leq g(\hat{u})\).

Key concepts

Continuity in Metric Spaces

When we talk about continuity in metric spaces, we're discussing a fundamental concept in real analysis. In a metric space, we can describe distances between points using a function known as a metric. The idea of continuity extends naturally to functions operating between metric spaces. A function is continuous at a point if small changes in the input near that point produce small changes in the output. Formally, for any positive number \(\epsilon\), however tiny, there exists another positive number \(\delta\) such that whenever we pinch the input within a \(\delta\)-neighborhood of a given point, the output will lie within an \(\epsilon\)-neighborhood of the function's value at that point.

For the function \(g(u)\) defined in our problem, its continuity means that as \(u\) edges closer to \(\bar{u}\) in the metric \(\rho\), the values of \(g(u)\) (which represent distances in the metric \(\sigma\)) also creep closer to \(g(\bar{u})\). Such behavior can indeed be baffling, but imagine it as a smooth transition with no unexpected leaps—an essential feature to ensure reliable predictions in various applications.

Uniform Continuity

Exploring uniform continuity unveils a more stringent form of continuity. It tells us that the function's output is not just behaving nicely around every single point, but rather across the entire domain. To claim a function is uniformly continuous, we ensure that for any small \(\epsilon\), there is a \(\delta\) that universally cures any two points in the domain—if they're within a \(\delta\)-distance, their images must be within an \(\epsilon\)-distance, no matter where we are in the space. It's like having a single rule of thumb for the whole space that prevents outputs from wildly fluctuating.

Uniform Continuity vs Standard Continuity

It's crucial to differentiate between uniform continuity and continuity at a point. While continuity at a point is localized, uniform continuity blankets the entire set uniformly with its \(\delta-\epsilon\) promise. Picture it as a net that gently cradles the entire graph of the function, ensuring it doesn't experience any sudden drops or spikes.

Compact Sets

Lastly, the significance of compact sets in the context of continuity cannot be overstated. In metric spaces, a set is called compact if every open cover has a finite subcover. To unravel this definition, envision an infinite number of open sets (like soft, infinite bubbles) that cover every point of our set. If you can always choose a finite number of these bubbles to still cover the set, then the set is compact. This concept is profound since compactness ensures some pleasant properties.

One remarkable aspect of compact sets is their 'closed and bounded' nature in Euclidean spaces, serving as a cornerstone for many theorems in analysis. In our exercise example, if we consider \(U\) to be compact, this implies that any continuous function defined on \(U\), such as \(g\), will not only be continuous but also uniformly continuous. Plus, the function will achieve its maximum and minimum values on \(U\). Essentially, compact sets help us tie down the 'wild' behavior of continuous functions, making them more manageable and predictable.