Question

(a) Suppose that \(y_{0}=\lim _{x \rightarrow x_{0}} g(x)\) exists and is an interior point of \(D_{f},\) and that \(f\) is continuous at \(y_{0} .\) Show that $$ \lim _{x \rightarrow x_{0}}(f \circ g)(x)=f\left(y_{0}\right) $$ (b) State an analogous result for limits from the right. (c) State an analogous result for limits from the left.

Short answer

Question: Prove the existence and equality of the limit of the composite function (f∘g)(x) as x approaches x0, given that f is continuous at y0, y0 is the limit of g(x) as x approaches x0, and y0 is an interior point of the domain Df. Short Answer: The existence of the limit and its equality to f(y0) are guaranteed because f is continuous at y0 and y0 is the limit of g(x) as x approaches x0. By using the continuity of f and the limit of g(x), we can find a δ for each ε > 0, such that the inequality |(f∘g)(x) - f(y0)| < ε holds for all x satisfying 0 < |x - x0| < δ. This proves the existence of the limit and its equality to f(y0).

Step-by-step solution

Setting up the given conditions

The given exercise states that: 1. \(y_{0}=\lim _{x \rightarrow x_{0}} g(x)\) 2. \(y_{0}\) is an interior point of domain \(D_{f}\) 3. \(f\) is continuous at \(y_{0}\) Now, let's show the required result: $$\lim _{x \rightarrow x_{0}}(f \circ g)(x)=f\left(y_{0}\right)$$

Prove the limit of the composite function

Given the property of continuity, since \(f\) is continuous at \(y_{0}\), for any \(\epsilon > 0\), there exists \(\delta_1 > 0\) such that: $$|f(y)-f(y_0)| < \epsilon \text{ whenever } |y-y_0|< \delta_1$$ Now, consider the limit of \(g(x)\) as \(x \rightarrow x_{0}\). There exists \(\delta_2 > 0\) such that, for any \(x\) with \(0 < |x-x_0|< \delta_2\), we have \(|g(x)-y_0|< \delta_1\). Let us choose \(\delta = \min(\delta_1, \delta_2)\). For any \(0 < |x-x_0|<\delta\), we have \(|g(x)-y_0|<\delta_1\). Then, for any \(y=g(x)\), we have \(|f(y)-f(y_0)|<\epsilon\), and that completes the proof.

Extending the result to right and left-hand limits

(b) Analogous result for limits from the right: If \(f\) is continuous at \(y_{0}\), \(g(x)\) converges to \(y_0\) from the right as \(x \rightarrow x_{0}^+\), that is, \(y_{0}=\lim _{x \rightarrow x_{0}^+} g(x)\). Then, we have $$\lim _{x \rightarrow x_{0}^+}(f \circ g)(x)=f\left(y_{0}\right)$$ (c) Analogous result for limits from the left: If \(f\) is continuous at \(y_{0}\), \(g(x)\) converges to \(y_0\) from the left as \(x \rightarrow x_{0}^-\), that is, \(y_{0}=\lim _{x \rightarrow x_{0}^-} g(x)\). Then, we have $$\lim _{x \rightarrow x_{0}^-}(f \circ g)(x)=f\left(y_{0}\right)$$ In summary, if \(y_{0}=\lim _{x \rightarrow x_{0}} g(x)\) exists and is an interior point of \(D_{f}\), and that \(f\) is continuous at \(y_{0}\), the result holds for all three cases: for general limits, right-hand limits, and left-hand limits.

Key concepts

Real Analysis

Real analysis is a branch of mathematical analysis dealing with the set of real numbers and real-valued functions. In the context of our textbook exercise, real analysis is concerned with understanding limits, continuity, and functions. When dealing with the limit of a composite function, real analysis provides the formal definitions and theorems that permit the evaluation of such limits, under appropriate conditions. This establishes the groundwork for proving sophisticated results, such as the behavior of functions as they approach specific points.

Part of mastering real analysis is being able to navigate the various kinds of limits and understanding the implications of a function's continuity. This skillset is essential for tackling more complex mathematical problems and is especially pertinent to the textbook exercise provided.

Continuity

Continuity is a fundamental concept in calculus and real analysis that describes a function's behavior around a point. A function is said to be continuous at a point if small changes in the input around that point result in small changes in the output. In simple terms, if you can draw the function without lifting your pencil from the paper, the function is continuous.

Regarding our exercise, continuity at the point \(y_0\) is crucial. Since \(f\) is continuous at \(y_0\), the values of \(f\) near \(y_0\) can be made as close as we like to \(f(y_0)\) by taking values of \(y\) that are sufficiently close to \(y_0\). This property is pivotal in establishing that the limit of the composition \((f \:circ \:g)(x)\) as \(x\) approaches \(x_0\) is the same as \(f\) evaluated at \(y_0\).

Interior Point

An interior point has a particular significance in real analysis and topology. It refers to a point within the domain of a function that has a ‘neighborhood’ completely contained within the domain. This neighborhood around the interior point allows one to move a little in any direction from this point without leaving the domain of the function.

In our exercise, \(y_0\) being an interior point of the domain \(D_f\) ensures that we can find other points near \(y_0\) where the function \(f\) is defined. It's one of the necessary conditions for applying the limit process to the composite function \(f\circ g\) and facilitates proving that the limit of the composite function as \(x\) approaches \(x_0\) is \(f(y_0)\).

Left-hand Limit

The left-hand limit is an important concept when discussing the limit of functions at a particular point. Specifically, it refers to what value a function approaches as the input comes arbitrarily close to a certain value from the left side. Mathematically, we denote the left-hand limit of \(g(x)\) as \(x\) approaches \(x_0\) by \(\lim_{x \rightarrow x_0^-} g(x)\).

Considering our exercise, if one needs to determine the left-hand limit of the composite function \(f \:circ \:g\), it would involve looking at the values of \(g(x)\) as \(x\) approaches \(x_0\) from the left, and then applying the continuous function \(f\) to those values. This limit is essential in analyzing the behavior of the composite function as it moves toward \(x_0\) from the left side.

Right-hand Limit

The counterpart of the left-hand limit is the right-hand limit. This is the value that a function is approaching as the input increases and gets ever closer to a given value from the right side. In notation, we express the right-hand limit of the function \(g(x)\) as \(x\) approaches \(x_0\) from the right by \(\lim_{x \rightarrow x_0^+} g(x)\).

In the scope of our exercise, the right-hand limit helps us understand the behavior of \(f \:circ \:g\) as \(x\) approaches \(x_0\) from the right. By establishing the continuity of \(f\) at \(y_0\) and using the fact that \(g(x)\) approaches \(y_0\) from the right, we can conclude that the right-hand limit of the composite function is equal to \(f(y_0)\), mirroring the approach used for general limits and left-hand limits.