Question

(a) Prove: If \(f: \mathbb{R} \rightarrow \mathbb{R}\) is continuous and locally invertible on \((a, b),\) then \(f\) is invertible on \((a, b)\). (b) Give an example showing that the continuity assumption is needed in (a).

Short answer

Question: Prove that if a function \(f: \mathbb{R} \rightarrow \mathbb{R}\) is continuous and locally invertible on an interval \((a, b)\), then it is invertible on the same interval. Give an example to show that if we remove the continuity assumption, the function may not be invertible on the given interval. Answer: If a function \(f: \mathbb{R} \rightarrow \mathbb{R}\) is continuous and locally invertible on an interval \((a, b)\), it is strictly monotonic (either strictly increasing or strictly decreasing) on that interval. This means it is both injective (one-to-one) and surjective (onto) on the interval, making it invertible on \((a, b)\). An example of a locally invertible function that is not continuous and therefore not invertible on any interval is: $$ f(x) = \begin{cases} x + 1, & \text{ if } x \in \mathbb{Q} \\ x - 1, & \text{ if } x \notin \mathbb{Q} \end{cases} $$ This function is not continuous and oscillates between increasing and decreasing in every neighborhood, making it not strictly monotonic and thus not invertible on any interval. This demonstrates the necessity of the continuity assumption.

Step-by-step solution

Prove that \(f\) is strictly monotonic

To prove that \(f\) is strictly monotonic, we need to show that it is either strictly increasing or strictly decreasing on the interval \((a, b)\). Since \(f\) is locally invertible on \((a, b)\), this means there exists a continuous local inverse function in a neighborhood around each point in \((a, b)\). Let's consider two arbitrary points \(x_1\) and \(x_2\) in \((a, b)\) such that \(x_1 < x_2\). Now, we can assume, without loss of generality, that \(f(x_1) < f(x_2)\). Since \(f\) is continuous and locally invertible, there exists a neighborhood \(U\) around \(x_1\) and a neighborhood \(V\) around \(x_2\) such that the function is invertible in these neighborhoods. Let \(g_1\) and \(g_2\) denote the inverse functions on \(U\) and \(V\), respectively. If we choose any point \(x_3 \in U\) such that \(x_1 < x_3 < x_2\), we have \(f(x_1) < f(x_3) < f(x_2)\). Similarly, if we choose any point \(x_4 \in V\) such that \(x_1 < x_4 < x_2\), we have \(f(x_1) < f(x_4) < f(x_2)\). As both \(x_3\) and \(x_4\) are arbitrary points in the neighborhoods of \(x_1\) and \(x_2\), we can conclude that \(f\) is strictly increasing on the interval \((a, b)\). Thus, we have proved that \(f\) is strictly monotonic (strictly increasing, in this case).

Conclude that \(f\) is invertible on \((a, b)\)

Since we have proved that \(f\) is strictly monotonic on the interval \((a, b)\), it must be injective (one-to-one) and surjective (onto). Therefore, \(f\) is invertible on \((a, b)\). For part (b):

Example without continuity assumption

Consider the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) defined as $$ f(x) = \begin{cases} x + 1, & \text{ if } x \in \mathbb{Q} \\ x - 1, & \text{ if } x \notin \mathbb{Q} \end{cases} $$ The function is locally invertible because, in any neighborhood \(U\) around any point \(x \in \mathbb{R}\), there are both rational and irrational numbers; so, the function has local inverses. However, the function \(f\) is not continuous, as the limit at any point \(x \in \mathbb{R}\) does not exist. Since the function oscillates between increasing and decreasing in every neighborhood and is therefore not strictly monotonic, it is not invertible on any interval. This example shows that the continuity assumption is needed in part (a).

Key concepts

Understanding Local Invertibility

When we talk about a function being locally invertible, it means that for every point in a particular interval, there exists a small neighborhood around that point where the function behaves like an invertible function. This means:

• It has a local inverse in the neighborhood, providing a way to "reverse" the function's effect within that small area.
• Local invertibility does not automatically imply that the function is invertible over a larger interval, unless certain conditions are met, such as strict monotonicity and continuity.

Imagine walking through a twisty corridor. At each step, there is a clear forward and backward path just around you. This is local invertibility. Still, navigating through the entire corridor smoothly from end to end relies on ensuring these local paths align properly (strictly monotonic) and without any unexpected steps (continuous). Without these additional properties, a globally smooth journey could not be guaranteed.

Strict Monotonicity and Its Role in Invertibility

Strict monotonicity is a crucial property when we're examining the invertibility of functions over an interval. A strictly monotonic function means that it is consistently increasing or consistently decreasing. Why does this matter?

• Being strictly increasing means for any two points, if one is larger, its image will also be larger.
• Being strictly decreasing is similar but in the opposite direction; a larger input will result in a smaller output.

Because of these behaviors, strict monotonicity ensures that a function is one-to-one (injective) and thus invertible on its entire interval. It's like knowing you're always gaining or losing height as you walk, ensuring no backtracking or repeating the same altitude, which would confuse your journey.

The Importance of Continuous Functions

Continuous functions play a significant role in ensuring that locally invertible functions become globally invertible over an interval. But what exactly is a continuous function? A function is continuous if you can draw it without lifting your pencil from the paper. In more formal terms:

• There are no sudden jumps or breaks at any point within the interval.
• The limit of the function at any point equals the function's value at that point.

In the realm of functioning behavior, continuity provides stability. It ensures that no unexpected changes disrupt the strict monotonic behavior required for global invertibility. Without continuity, a function can behave erratically, as shown in the example presented in the solution, where the function dealt differently with rational and irrational numbers. Here, despite being locally invertible, lack of continuity led to chaos, proving it non-invertible globally. Through continuity, we secure a smooth, predictable path across the entire interval, vital for ensuring comprehensive invertibility.