Let D ⊆ℝ be symmetric about the origin, that is, -x ∈D whenever x ∈D . If c ∈D and f: D →ℝ is ei

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Let D ⊆ℝ be symmetric about the origin, that is, -x ∈D...

Let $D \subseteq \mathbb{R}$ be symmetric about the origin, that is, $-x \in D$ whenever $x \in D .$ If $c \in D$ and $f: D \rightarrow \mathbb{R}$ is either an even or an odd function, then show that the left (hand) derivative $f_{-}^{\prime}(c)$ at $c$ exists if and only if the right (hand) derivative $f_{+}^{\prime}(-c)$ at $-c$ exists. Further, if either (and hence both) of these derivatives exists, then show that $f_{-}^{\prime}(c)=-f_{+}^{\prime}(-c)$ if $f$ is even, and $f_{-}^{\prime}(c)=f_{+}^{\prime}(-c)$ if $f$ is odd. Deduce that if $f$ is differentiable, then $f^{\prime}$ is an odd (resp. even) function according as $f$ is an even (resp. odd) function.

Key Concepts

Symmetric Domains

A symmetric domain in mathematics is a set in which, whenever an element x is included, its negation -x is also included. This concept is crucial in analyses involving even and odd functions because it allows one to relate the behavior of the function at opposite points, which is often exploited in proofs regarding derivatives and integrals in symmetric regions.

Even Functions

Even functions are those functions that satisfy the property f(x) = f(-x) for all x in the domain. This symmetry means that the graph of an even function is mirrored across the y-axis, and it plays a significant role in determining the relationships between derivatives at symmetric points, particularly when comparing left and right hand derivatives.

Odd Functions

Odd functions are defined by the property f(-x) = -f(x) for all x in the domain. This means their graphs are symmetric with respect to the origin, providing a neural symmetry that is useful in analysis, especially when evaluating derivative behaviors at opposite points or when integrating over symmetric intervals.

One-sided Derivatives

One-sided derivatives, including left-hand and right-hand derivatives, are defined as the limits of the difference quotient as the variable approaches a point from the left or right, respectively. These concepts are fundamental in understanding differentiability at points where the function may behave differently when approached from different directions, particularly in the context of functions defined on symmetric intervals.

Limit Definition of the Derivative

The derivative of a function at a point is fundamentally defined as the limit of the difference quotient as the increment approaches zero. This limit-based approach underscores the importance of considering one-sided limits in cases of non-smooth behavior and is critical when linking the properties of even or odd functions with their derivative behavior at symmetric points.

Differentiability and Parity of the Derivative

Differentiability refers to a function having a derivative at every point in its domain. In the context of even and odd functions, a key result is that if an even or odd function is differentiable, then its derivative exhibits an opposite symmetry: the derivative of an even function is odd, and the derivative of an odd function is even. This relationship leverages the underlying symmetry of the original function, along with the limit definitions of derivatives.

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