Consider the function g: ℝ \{0} →ℝ defined by g(x):=cos(1 / x) . Prove the following: (i) g is a

step 1 So we know that according to the laws of continuity, any basic functions, which the absolute val

Consider the function g: ℝ \{0} →ℝ defined by g(x):=cos(1 /...

Consider the function $g: \mathbb{R} \backslash\{0\} \rightarrow \mathbb{R}$ defined by $g(x):=\cos (1 / x) .$ Prove the following: (i) $g$ is an even function. (ii) $\lim _{x \rightarrow 0} g(x)$ does not exist, but $\lim _{x \rightarrow 0^{+}}[g(x)-g(-x)]$ exists. Also, $g$ cannot be extended to $\mathbb{R}$ as a continuous function. (iii) For any $\delta>0, g$ is not uniformly continuous on $(0, \delta)$ as well as on $(-\delta, 0)$, but it is uniformly continuous on $(\infty,-\delta] \cup[\delta, \infty)$. (iv) For any $\delta>0, g$ is not monotonic, not convex, and not concave on $(0, \delta)$ as well as on $(-\delta, 0)$.

Consider the function $g: \mathbb{R} \backslash\{0\} \rightarrow \mathbb{R}$ defined by $g(x):=\cos (1 / x) .$ Prove the following:
(i) $g$ is an even function.
(ii) $\lim _{x \rightarrow 0} g(x)$ does not exist, but $\lim _{x \rightarrow 0^{+}}[g(x)-g(-x)]$ exists. Also, $g$
cannot be extended to $\mathbb{R}$ as a continuous function.
(iii) For any $\delta>0, g$ is not uniformly continuous on $(0, \delta)$ as well as on $(-\delta, 0)$, but it is uniformly continuous on $(\infty,-\delta] \cup[\delta, \infty)$.
(iv) For any $\delta>0, g$ is not monotonic, not convex, and not concave on $(0, \delta)$ as well as on $(-\delta, 0)$.

Key Concepts

Even Functions

An even function is one that satisfies the condition f(?x) = f(x) for all x in its domain. This symmetry about the y?axis is a fundamental property in analysis, allowing simplifications when studying the behavior of functions, their integrals, and their graphs.

Limits and One-Sided Limits

The concept of a limit investigates the behavior of a function as the variable approaches a particular point, often leading to an understanding of continuity or discontinuity at that point. In cases where a function exhibits oscillatory or other erratic behavior, the overall limit may not exist despite the possibility of the existence of one-sided limits, which examine the behavior only from one direction.

Oscillatory Behavior and Non-Extendability

Oscillatory functions are those that do not settle to a single value as their input approaches a particular point, often due to high-frequency oscillations. This behavior is key in understanding why some functions cannot be continuously extended to include certain limit points, despite well-defined behavior on either side of the point.

Uniform Continuity

Uniform continuity strengthens the standard concept of continuity by requiring that the function's rate of change remains controlled across the entire domain, not just near a specific point. This concept is critical when comparing behavior over different subsets of the domain, especially in distinguishing regions where the function is uniformly continuous from regions where fast oscillations or steep gradients prevent uniform behavior.

Monotonicity

A function is monotonic on an interval if it is either entirely non-decreasing or non-increasing throughout that interval. Monotonicity is an important property because it simplifies analysis by ensuring that functions have no local extrema within that domain, and it is often related to conditions that guarantee other properties like continuity or integrability.

Convexity and Concavity

Convexity and concavity are properties describing the curvature of a function. Convex functions have graphs that lie below the line segment connecting any two points in the domain, whereas concave functions lie above such segments. These concepts are central in optimization and economic theory, and the absence of convexity or concavity can indicate complex behavior such as multiple local extrema or oscillations.

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