Question

Suppose that \(f\) and \(g\) are defined on \((-\infty, \infty), f\) is increasing, and \(f \circ g\) is continuous on \((-\infty, \infty)\). Show that \(g\) is continuous on \((-\infty, \infty)\).

Short answer

Question: Given \(f\) and \(g\) are defined on \((-\infty, \infty)\), \(f\) is increasing, and \(f \circ g\) is continuous on \((-\infty, \infty)\), prove that \(g\) is also continuous on \((-\infty, \infty)\). Answer: Since the limit of \(f(g(x))\) as \(x \to x_0\) exists and \(g(x)\) is squeezed between the increasing function \(f\) (or its inverse function \(f^{-1}\)) and \(g(x_0)\), the limit of \(g(x)\) as \(x \to x_0\) must also exist, making \(g\) continuous on \((-\infty, \infty)\).

Step-by-step solution

Define the problem

We are given that \(f\) and \(g\) are defined on \((-\infty, \infty)\), \(f\) is increasing, and \(f \circ g\) is continuous on \((-\infty, \infty)\). Our goal is to show that \(g\) is continuous on \((-\infty, \infty)\).

Write the definition of continuity

A function is continuous at a point \(x_0\) if \(\lim_{x \to x_0} g(x) = g(x_0)\). We want to show that this is true for \(g\) for any \(x_0\) in \((-\infty, \infty)\).

Use the fact that \(f \circ g\) is continuous

Since \(f \circ g\) is continuous for all \(x \in (-\infty, \infty)\), we know that \(\lim_{x \to x_0} (f \circ g)(x) = (f \circ g)(x_0)\).

Write the limit using the composite function

We can write the limit as \(\lim_{x \to x_0} f(g(x)) = f(g(x_0))\).

Apply the increasing nature of \(f\)

Since \(f\) is increasing, we know that if \(g(x) < g(x_0)\), then \(f(g(x)) < f(g(x_0))\), and if \(g(x) > g(x_0)\), then \(f(g(x)) > f(g(x_0))\).

Use the squeeze theorem to find the limit of \(g(x)\)

Combining the results of Steps 4 and 5, we know that: \begin{eqnarray*} g(x) < g(x_0) &\Rightarrow& f(g(x)) < f(g(x_0)), \\ g(x) > g(x_0) &\Rightarrow& f(g(x)) > f(g(x_0)). \end{eqnarray*} Since the limit of \(f(g(x))\) as \(x \to x_0\) exists, and \(g(x)\) is squeezed between the increasing function \(f\) (or its inverse function \(f^{-1}\)) and \(g(x_0)\), the limit of \(g(x)\) as \(x \to x_0\) must also exist. Hence, we have \(\lim_{x \to x_0} g(x) = g(x_0)\).

Conclusion

We have shown that for any point \(x_0 \in (-\infty, \infty)\), the limit \(\lim_{x \to x_0} g(x)\) exists and is equal to \(g(x_0)\). Therefore, \(g\) is continuous on \((-\infty, \infty)\).

Key concepts

Understanding Function Composition

When studying mathematical functions, one critical concept is function composition, which involves applying one function to the results of another. Imagine you have two functions, f and g, where each takes a number and transforms it in some way. Function composition is the process of combining these two transformations into one by applying g first and then f. In formal terms, this is denoted as f following g, or f \( \circ \) g, and is defined as f(g(x)). This becomes crucial when we want to understand how combining functions affects properties like continuity, which brings us to another fundamental concept.

Definition of Continuity

Continuity is a property of functions that mathematicians treasure for its smoothness and predictability. For a function to be continuous at a particular point, three conditions must be met: the function must be defined at the point, the limit of the function must exist as you approach the point, and both the function's value and the limit must be equal at that point. In layman's terms, you can think of a continuous function as one where you can draw its graph without lifting your pencil off the paper. In mathematical notation, for a function g to be continuous at a point x_0, this is captured by the limit: \(\lim_{x \to x_0} g(x) = g(x_0)\).
Understanding this definition is key to solving problems where you need to prove continuity, such as the exercise we're working with, where we want to show that a function g is continuous across an entire interval.

Increasing Functions

In real analysis, functions can behave like hikers trekking up a mountain—they can either go up or down, or even remain level. A function is termed increasing if, as you move from left to right along the x-axis, the y-value consistently rises. This doesn't necessarily mean the function can't stay flat for a while—it can, and this is a weakly increasing function. However, if it continually goes up without any flat segments, it's called strictly increasing. Mathematically speaking, for a function f and any two numbers in its domain such that a < b, an increasing function will satisfy f(a) ≤ f(b), and a strictly increasing function will satisfy f(a) < f(b). The fact that function f in our exercise is increasing is significant because it imposes an order that we can use to better understand the behavior of the composed function f \( \circ \) g.

Applying the Squeeze Theorem

Imagine you're squeezing a balloon between your hands. No matter how the balloon flexes, its edges are constrained by your palms. This is similar to how the Squeeze Theorem works in mathematics. The theorem lets us calculate the limit of a function that's 'squeezed' between two other functions that meet at a common limit. To use this theorem, we need three functions where two (let's call them f and h) are easy to handle, and they trap the third function g in-between. If as we approach a point x_0, both f and h approach the same limit, then g is forced to do the same.
This theorem is particularly handy when dealing with continuous functions and is used in the exercise to argue that since the limit of f(g(x)) exists and f is increasing, g(x) must be squeezing its way to continuity as x approaches x_0. In this way, the more complex behavior of g is 'controlled' by the simpler, increasing function f, assuring us of g's continuity.