Question

Prove that if \(f\) is differentiable at \(p\) then \(\lim _{x \rightarrow p^{+}} \frac{f(x)-f(y)}{x-y}\) exists, is finite, and equals \(f^{\prime}(p)\) $$ \begin{array}{l} \text { i.e., }(\forall \varepsilon>0)(\exists \delta>0)(\forall x \in(p, p+\delta))(\forall y \in(p-\delta, p)) \\ \left|\frac{f(x)-f(y)}{x-y}-f^{\prime}(p)\right|<\varepsilon \end{array} $$ Show, by redefining \(f\) at \(p,\) that even if the limit exists, \(f\) may not be differentiable (note that the above limit does not involve \(f(p))\).

Short answer

The proof shows limit exists at point \( p \) by continuity, but redefining \( f(p) \) can break differentiability.

Step-by-step solution

Understand Differentiability at a Point

A function \( f \) is differentiable at a point \( p \) if the limit \( \lim_{x \to p} \frac{f(x) - f(p)}{x - p} \) exists. This means that there is a finite derivative \( f'(p) \) at that point.

Establish the Right-Hand Limit

We need to show that the limit \( \lim_{x \to p^+} \frac{f(x) - f(y)}{x - y} \) exists and equals \( f'(p) \). For when \( x > p \) and \( y < p \), use the definition of differentiability:\( f'(p) = \lim_{x \to p} \frac{f(x) - f(p)}{x - p} \).Note that as \( x \to p^+ \), both function increments \( f(x) - f(p) \) and \( f(y) - f(p) \) behave similarly if \( f \) is continuous around \( p \).

Bounding the Expression

By the definition of differentiability and continuity, \( \lim_{x \to p^+} f(x) = f(p) \) and \( \lim_{y \to p^-} f(y) = f(p) \).Therefore, we can manipulate \( \frac{f(x) - f(y)}{x - y} = \frac{(f(x) - f(p)) - (f(y) - f(p))}{x - y} \).For small \( \delta > 0 \), both terms can be expressed in terms of the derivative \( f'(p) \).

Evaluating the Limit

As both \( x \to p^+ \) and \( y \to p^- \), both terms \( \frac{f(x) - f(p)}{x - p} \) and \( \frac{f(y) - f(p)}{y - p} \) tend to \( f'(p) \). By the difference quotient properties:\( \left|\frac{f(x)-f(y)}{x-y} - f'(p)\right| = \frac{1}{x-y} (f(x) - f(y)) - f'(p) \rightarrow 0. \)This satisfies the epsilon-delta condition given in the problem, proving the first statement.

Redefine Function at a Point to Show Limit Existence Without Differentiability

To show that differentiability is not guaranteed, consider a piecewise function:\[ f(p) = different\ value \ and \ f(x) \ for \ x eq p \] If the limit exists due the continuity around \( p \) without the involvement of \( f(p) \), such definition could break differentiability at \( p \) while still having the right-hand limit equal \( f'(p) \).

Concluding Example

Consider a function defined by:- \( f(x) = x^2 \) for \( x eq p \)- \( f(p) = k \) where \( k eq p^2 \).The limit \( \lim_{x \to p^+} \frac{f(x)-f(y)}{x-y} = 2p \) exists and is finite, but the function is not differentiable at \( p \) due to the break in the function definition from \( f(p) \).

Key concepts

Limit of a function

The concept of a limit is fundamental in calculus. It helps us understand the behavior of a function as the input approaches a particular value. When we say the limit of a function exists at a certain point, it means that as the variable gets closer to that point, the function's output gets closer to a specific value.
For a function \( f \) as \( x \to p \), the limit is noted as \( \lim_{x \to p} f(x) \). If this limit results in a single real number, the function is considered to have a limit at that point. This provides a basis for further concepts like continuity and differentiability.

Importance in Calculus

Limits bridge the gap between algebra and calculus and provide a tool for analyzing points where functions may not be explicitly defined or predictable. Without limits, concepts like derivatives and continuity would be hard to define.

• Provides a way to explore function behavior at points of interest.
• Foundational for understanding derivatives, which are limits in themselves.

Epsilon-delta definition

The epsilon-delta definition is a rigorous way of defining the limit of a function. It's a formal mathematical approach used to demonstrate that as one quantity (\(x\)) gets arbitrarily close to another quantity (\(p\)), another quantity (\(f(x)\)) approaches a specific value \(L\).
More precisely, \( \lim_{x \to p} f(x) = L \) if for every \( \varepsilon > 0 \), there exists a \( \delta > 0 \) such that whenever \( 0 < |x - p| < \delta \), it follows that \( |f(x) - L| < \varepsilon \). This definition is crucial in understanding the behavior of functions around specific points.

Key Points of the Definition

The epsilon-delta definition assures us of the proximity of function values to \(L\) based on how close \(x\) is to \(p\). This ensures mathematical rigor in proofs concerning limits and provides a way to verify or disprove the existence of a limit.

• \(\varepsilon\) represents how close we want \(f(x)\) to be to \(L\).
• \(\delta\) indicates how close \(x\) needs to be to \(p\).
• The definition applies universally, making it a powerful tool for analyzing function behavior.

Right-hand limit

The right-hand limit of a function is concerned with the behavior of the function as the variable approaches a specific point from the right, or from values greater than the point. It's denoted as \( \lim_{x \to p^+} f(x) \).
Understanding right-hand limits is essential when dealing with piecewise functions or when considering function behavior on specific intervals. For instance, in a differentiability problem, the right-hand limit can help determine how the function approaches a particular derivative value as \(x\) approaches from one side.

Significance in Analysis

Right-hand limits are particularly useful in cases where a function's behavior might be different on either side of a point.

• Helpful when examining piecewise functions where behavior might differ around certain points.
• Crucial in analyzing one-sided derivatives and limits.
• Often used in conjunction with left-hand limits to assess full directional continuity or differentiability.

Piecewise function

A piecewise function is a function that is defined by different expressions over different intervals. These types of functions are common in calculus and real-world applications, as they can model situations where a rule changes based on different conditions or inputs.
They are written with each piece in separate segments and show the function expressions that apply over specified domains. An example might be a tax bracket system or a physics problem with changing velocity or force over time.

Characteristics of Piecewise Functions

Piecewise functions can be uniquely suited for solving differential problems where behavior changes at certain points.

• Allows for multiple rules within a single function framework.
• Useful in real-world modeling where conditions are not uniform.
• Can influence the differentiability of a function at the boundaries of the pieces.

Knowing how to work with piecewise functions is crucial because they can often have limits and derivatives at points where the rules change. In differentiation, this can mean checking each segment for continuity and differentiability.