Question

(i) Let \(f:(a, b) \rightarrow E\) be finite, continuous, with a right derivative on \((a, b) .\) Prove that \(q=\lim _{x \rightarrow a^{+}} f_{+}^{\prime}(x)\) exists (finite) iff $$ q=\lim _{x, y \rightarrow a^{+}} \frac{f(x)-f(y)}{x-y} $$ i.e., iff $$ (\forall \varepsilon>0)(\exists c>a)(\forall x, y \in(a, c) \mid x \neq y) \quad\left|\frac{f(x)-f(y)}{x-y}-q\right|<\varepsilon $$ [Hints: If so, let \(y \rightarrow x^{+}\) (keeping \(x\) fixed) to obtain $$ (\forall x \in(a, c)) \quad\left|f_{+}^{\prime}(x)-q\right| \leq \varepsilon . \quad \text { (Why?) } $$ Conversely, if \(\lim _{x \rightarrow a^{+}} f_{+}^{\prime}(x)=q,\) then $$ (\forall \varepsilon>0)(\exists c>a)(\forall t \in(a, c)) \quad\left|f_{+}^{\prime}(t)-q\right|<\varepsilon $$ Put $$ M=\sup _{a

Short answer

The limit of the right-hand derivative exists finite if and only if this equivalence of limits regarding derivatives and incremental ratios holds.

Step-by-step solution

Understanding the Problem

We are given a function \( f:(a, b) \rightarrow E \) that is finite, continuous, and has a right derivative on \((a, b)\). We need to prove the equivalence of two limits regarding the derivatives and the incremental ratios of \( f \).

Establishing Forward Direction

Suppose \( q=\lim_{x \rightarrow a^{+}} f_{+}^{\prime}(x) \) exists finite. To show,\[\lim_{x, y \rightarrow a^{+}} \frac{f(x)-f(y)}{x-y} = q\]we use the hint and let \( y \rightarrow x^+ \) with \( x \) fixed in \((a, c)\), the working domain. Then, the right-hand derivative being close to \( q \) implies that the increment difference also tends to \( q \), thus proving the forward direction.

Establishing Converse Direction

Conversely, assume that \[\lim_{x, y \rightarrow a^{+}} \frac{f(x)-f(y)}{x-y} = q\]exists. For any \( \varepsilon > 0 \), there exists \( c > a \) such that for all \( x, y \in (a, c) \), \[ \left|\frac{f(x)-f(y)}{x-y} - q\right| < \varepsilon.\] By choosing \( y \to x^+ \), this implies \[ \left| f_+^{\prime}(x) - q \right| < \varepsilon, \]showing that \( q = \lim_{x \rightarrow a^{+}} f_+^{\prime}(x) \).

Verification with Function Attenuation

Define \( h(t) = f(t) - tq \). Based on provided hints and using Corollary 1, for \([x, y] \subseteq (a, c)\), we have:\[ |f(y)-f(x)-(y-x)q| \leq M(y-x) \leq \varepsilon(y-x) \] from which the limit equivalence can be derived, assuring both directions.

Similar Statements for Infinity

For \( q = \pm \infty \) or \( x \to b^- \), follow hints in Problem 7(ii) by comparing increment ratios with the derivative approaching infinity. Use scaling arguments to see that the inverse of increment approximates the rate of growth or shrink.

Key concepts

Limit of Derivative

When dealing with functions that are continuous and have right derivatives, we often explore the **limit of the derivative** as a way of understanding how the function behaves near a certain point. Here, we are concerned with the right derivative of a function as we approach a point from the right side of the interval. The expression \(\lim_{x \rightarrow a^+} f_{+}^{\prime}(x) = q\) suggests that as \(x\) approaches \(a\) from the right, the value of the right derivative \(f_{+}^{\prime}(x)\) gets closer and closer to \(q\).

This concept is vital because it provides a precise way to describe the behavior of the function's slope at the edge of the interval. The limit must be finite, meaning the derivative doesn't shoot off to infinity, ensuring the function is not making any abrupt changes at that point. If \(q\) exists and is finite, it implies that the derivatives are approaching a specific constant value, leading to a "smooth" transition as we pass through that point.

Continuity of Functions

Understanding the **continuity of functions** is crucial when working with derivatives. A function is continuous on an interval if, informally, you can draw it without lifting your pencil from the paper. For a function \(f\) in our context, it means there are no holes, jumps, or vertical asymptotes in the interval \((a, b)\).

The property of continuity ensures that the limit of the function's derivatives (whether from the left or right) approaches the same value. It contributes significantly to the end behaviour of the function as described by its derivatives. For example, the certainty that \(f\) is continuous rightward from \(a\) allows us to work with the right derivatives and their limits, knowing that the behavior of \(f\) is predictable based on the derivative information alone without sudden changes.

Incremental Ratio

The **incremental ratio** is a core concept related to the average rate of change of the function over an interval. It is given by the expression \(\frac{f(x)-f(y)}{x-y}\) for the function \(f\) defined over the interval \((a, b)\). This computation helps us understand how much \(f\) changes between two points, \(x\) and \(y\), within the interval.

The statement involving limits of this ratio as \(x, y \to a^+\) informs us about the right derivative's behaviour near \(a\). When this limit exists and equals some \(q\), it confirms that the function's instantaneous rate of change (derivative) at \(a\) matches this average rate of change, making it a key aspect for calculus students to grasp how instantaneous and average rates interplay.

Directional Limits

Directional limits are integral when evaluating **limit-of-derivative scenarios**. They help detect the tendencies of a function as it approaches a point from one side (right or left) in the interval. For instance, in this problem, we use the notation \(x \to a^+\) to express approaching \(a\) from the right.

Such directional concepts help ensure that the right derivative, as we approach \(a\), converges to the same value \(q\) established by the incremental ratio. Controlling these limits is critical in confirming not only the existence of \(q\) but also in verifying our function's continuity, especially around edges of defined intervals where sudden functional transitions can otherwise occur.
Understanding directional limits grants us comprehensive insights into convergence behaviour, central to continuous function analysis and derivation.