Question

Use Theorem 2.2 .7 to tind all points \(x_{0}\) at which the following functions are continuous. (a) \(\sqrt{1-x^{2}}\) (b) \(\sin e^{-x^{2}}\) (c) \(\log (1+\sin x)\) (d) \(e^{-1 /(1-2 x)}\) (e) \(\sin \frac{1}{(x-1)^{2}}\) (f) \(\sin \left(\frac{1}{\cos x}\right)\) (g) \(\left(1-\sin ^{2} x\right)^{-1 / 2}\) (h) \(\cot \left(1-e^{-x^{2}}\right)\) (i) \(\cos \frac{1}{x}\)

Short answer

Using Theorem 2.2.7, determine the points of continuity for each of the following functions: (a) \(\sqrt{1-x^2}\) (b) \(\sin e^{-x^{2}}\) (c) \(\log (1+\sin x)\) (d) \(e^{-1 /(1-2 x)}\) (e) \(\sin \frac{1}{(x-1)^{2}}\) (f) \(\sin \left(\frac{1}{\cos x}\right)\) (g) \(\left(1-\sin ^{2} x\right)^{-1 / 2}\) (h) \(\cot \left(1-e^{-x^{2}}\right)\) (i) \(\cos \frac{1}{x}\) For each function, determine its points of continuity as follows: (a) Continuous for \(x\in [-1,1]\) (b) Continuous for all \(x\) (c) Continuous for all \(x\) (d) Continuous for all \(x\) except when \(x=\frac{1}{2}\) (e) Continuous for all \(x\) except when \(x=1\) (f) Continuous for all \(x\) except for integer multiples of \(\frac{\pi}{2}\) (g) Continuous for all \(x\) (h) Continuous everywhere except when \(1-e^{-x^2} = k\pi\) with \(k \in \mathbb{Z}\) (i) Continuous for all \(x\) except when \(x=0\)

Step-by-step solution

Continuity of \(\sqrt{1-x^2}\)

This function is continuous where the inner expression \((1-x^2)\) is non-negative, as the square root is continuous for non-negative numbers. Hence, the function is continuous for \(x\in [-1,1]\). (b)

Continuity of \(\sin e^{-x^2}\)

We know that the sine function is continuous everywhere, as is the exponential function. Since both the sine and the exponential functions are continuous, their composition is also continuous. Thus, this function is continuous for all \(x\). (c)

Continuity of \(\log(1+\sin x)\)

The function \(\log(x)\) is continuous for positive x values. Additionally, \(\sin x\) varies between -1 and 1. Therefore, the domain of the function is \([-\pi, \pi]\), making the function continuous for all \(x\). (d)

Continuity of \(e^{-1 /(1-2 x)}\)

The exponential function is continuous everywhere. However, the problem arises when the denominator of the exponent, \((1-2x)\), is equal to 0. In this case, it is undefined. This is the only discontinuity, which occurs when \(x=\frac{1}{2}\). Therefore, the function is continuous for all \(x\) except when \(x=\frac{1}{2}\). (e)

Continuity of \(\sin \frac{1}{(x-1)^{2}}\)

The sine function is continuous everywhere. Therefore, the only condition is the continuity of the function \(\frac{1}{(x-1)^{2}}\). This function is continuous everywhere except when \(x=1\). Hence, the function is continuous for all \(x\) except when \(x=1\). (f)

Continuity of \(\sin \left(\frac{1}{\cos x}\right)\)

The sine function is continuous everywhere. However, the cosine function is discontinuous when its value is 0. Consequently, continuous points occur when \(\cos x \neq 0\). This function is continuous at all points \(x\) except for integer multiples of \(\frac{\pi}{2}\). (g)

Continuity of \(\left(1-\sin ^{2} x\right)^{-1 / 2}\)

This function is continuous where the inner expression \((1-\sin ^{2} x)\) is non-negative and non-zero (due to the exponent being -1/2). Since \(\sin^2 x\) lies between 0 and 1, this function is continuous for all \(x\). (h)

Continuity of \(\cot \left(1-e^{-x^{2}}\right)\)

The cotangent function has a discontinuity when its argument is an integer multiple of \(\pi\). Moreover, since \(e^{-x^2}\) lies between 0 and 1, the discontinuities occur when \(1-e^{-x^2} = k\pi\) with \(k \in \mathbb{Z}\). The function is continuous everywhere else. (i)

Continuity of \(\cos \frac{1}{x}\)

The cosine function is continuous everywhere. However, it is undefined when \(x=0\), which results in a discontinuity. Therefore, the function is continuous for all \(x\) except when \(x=0\).

Key concepts

Real Analysis

Real analysis is a branch of mathematical study that deals with the properties and behavior of real numbers and real-valued functions. It lays the foundation for understanding concepts such as limits, continuity, differentiation, and integration - all of which are essential for rigorous mathematical reasoning and application.

When discussing continuity within real analysis, it is crucial to examine the behavior of functions around specific points in their domain. A function is said to be continuous at a point if, intuitively speaking, you can draw its graph without lifting your pencil, which means the function does not have breaks, jumps, or points of discontinuity at that point.

For a function to be continuous at a point, it must satisfy three conditions: it must be defined at that point; its limit as it approaches that point must exist; and the limit must equal the function's value at that point. Real analysis rigorously defines these intuitions, providing a clear framework for understanding how functions behave and interact.

Theorem Application

The application of theorems in analyzing the continuity of functions is a fundamental technique in real analysis. Theorems give us rules and conditions under which certain mathematical statements are true, which we can then use to solve problems or prove additional statements.

One such theorem mentioned in the exercise is Theorem 2.2.7, which likely states conditions for continuity of functions (though the precise content is not provided in the example). Applying this theorem allows us to systematically identify points of continuity for a variety of functions, such as those involving square roots, trigonometric functions, exponentials, and logarithms.

Utilizing the theorems involves identifying the domain of the function and understanding the behavior of the function's components (like sine, cosine, exponential, etc.). Then, you evaluate the continuity based on the function's definition and the theorem's criteria. This application enables us to conclude, for instance, that the function \(\sin e^{-x^2}\) is continuous for all real numbers, considering that both the sine function and the exponential function are individually continuous everywhere.

Continuous Function Properties

Properties of continuous functions are central to understanding how they behave across their domains. These properties are not just theoretical; they have practical implications in fields such as engineering, physics, and economics.

One key property is that continuous functions transform small changes in their input into small changes in their output, which is essential in the context of stability and sensitivity analysis in real-world applications. Another is that they maintain the intermediate values: if a continuous function takes on two values at different points in its domain, then it must take on every value in between these two at some point in the domain - this is known as the Intermediate Value Theorem.

In our exercise, the continuity property helps us determine that the function \(e^{-1 /(1-2 x)}\) is continuous everywhere except at \(x=\frac{1}{2}\), since at that point the denominator of the exponent becomes zero, causing undefined behavior in the function. Understanding these properties is not only crucial for solving textbook problems but also for modeling and interpreting complex real-world situations.