Question

Show that \(f(a+)\) and \(f(b-)\) exist (finite) if \(f^{\prime}\) is bounded on \((a, b)\). HINT: See Exercise \(2.1 .38 .\)

Short answer

Question: Show that the right-hand limit \(f(a+)\) and the left-hand limit \(f(b-)\) exist and are finite if the derivative \(f^{\prime}\) is bounded on the open interval \((a, b)\). Answer: If the derivative \(f^{\prime}\) is bounded on the open interval \((a, b)\), then the right-hand limit \(f(a+)\) and the left-hand limit \(f(b-)\) exist and are finite as shown in the step-by-step solution. This holds due to the application of Exercise 2.1.38 and the definition of the right-hand and left-hand limits.

Step-by-step solution

Recall the definitions of the right-hand and left-hand limits

The right-hand limit of a function \(f(x)\) at a point \(a\) is defined as: $$ f(a+) = \lim_{x \to a+} f(x) $$ Similarly, the left-hand limit of a function \(f(x)\) at a point \(b\) is defined as: $$ f(b-) = \lim_{x \to b-} f(x) $$

Use Exercise 2.1.38 to find expressions for \(f(a+)\) and \(f(b-)\)

According to Exercise 2.1.38, if a function \(F\) is increasing and differentiable on the open interval \((c, d)\) and has a finite derivative \(F'\), then the limit of \(F\)'s derivative when \(x\) approaches \(c\) from the right exists and is equal to \(F(c+)\): $$ \lim_{x \to c+} F(x) = F(c+) $$ Let \(F(x) = f(x) - f(a) - M(x-a)\), where \(M\) is the bound for \(f^{\prime}\) on \((a, b)\). \(F(x)\) is differentiable on \((a, b)\), and its derivative is \(F^{\prime}(x) = f^{\prime}(x) - M\), which is bounded on \((a, b)\) by hypothesis. Since \(F^{\prime}(x)\) is non-negative on \((a, b)\), \(F(x)\) is increasing on this interval. Applying Exercise 2.1.38, we have: $$ \lim_{x \to a+} F(x) = F(a+) = f(a+) - f(a) - M(a+ - a) $$ and $$ \lim_{x \to b-} F(x) = F(b-) = f(b-) - f(a) - M(b- - a) $$

Prove that \(f(a+)\) and \(f(b-)\) are finite

Since \(f(a)\) and \(M\) are finite, we can rewrite the expressions for \(f(a+)\) and \(f(b-)\) as: $$ f(a+) = f(a) + M(a+ - a) + F(a+) $$ and $$ f(b-) = f(a) + M(b- - a) + F(b-) $$ Knowing that \(F(a+)\) and \(F(b-)\) are finite, we can now conclude that \(f(a+)\) and \(f(b-)\) are finite as well. Therefore, we have shown that \(f(a+)\) and \(f(b-)\) exist and are finite if \(f^{\prime}\) is bounded on \((a, b)\).

Key concepts

Bounded Derivative

A bounded derivative is a key concept in calculus, referring to a function whose derivative does not exceed a certain limit in magnitude over a given interval. In simpler terms, if you have a function, and its rate of change—how quickly it is increasing or decreasing—stays within certain limits, the derivative is considered bounded.
This property is essential in proving the existence and finiteness of key limits, like the right-hand and left-hand limits.

Why is this important?

• Bounded derivatives ensure the function behaves in a controlled, predictable manner.
• They prevent the function from having steep inclines or wild oscillations that would cause issues with continuity and limit calculations.

Whenever you're dealing with functions on specific intervals, checking if the derivative is bounded can vastly simplify your analysis and provide insights into the function's behavior.

Right-Hand Limit

The right-hand limit of a function at a given point tells us how the function behaves as the input approaches the point from the right. In mathematical terms, for a function \( f(x) \) at the point \( a \), this is represented as \( f(a+) = \lim_{x \to a+} f(x) \).
The right-hand limit is crucial when dealing with functions that might have jumps or discontinuities at certain points, as it helps characterize the behavior just before these possible "jumps."

Important aspects:

• It is primarily used when a function is not necessarily continuous across a whole interval but is instead being analyzed at the borders of sub-intervals.
• A finite right-hand limit indicates that the function does not spiral off to infinity right as it hits a tipping point from the right.

Understanding right-hand limits helps in understanding complex functions more deeply, particularly those that change behavior at specific places.

Left-Hand Limit

The left-hand limit of a function is the concept used to understand how the function behaves as the input approaches a given point from the left. Mathematically, it’s denoted as \( f(b-) = \lim_{x \to b-} f(x) \) for some point \( b \).
Left-hand limits are beneficial for analyzing points where functions might have abrupt changes, akin to right-hand limits.

Points to remember:

• They allow insight into the function’s behavior just as it reaches a point from the left.
• If the left-hand limit is finite, it suggests a controlled behavior of the function rather than drastic oscillations or undefined behavior.

Using left-hand and right-hand limits together, you can get a well-rounded view of the function's behavior around its endpoints or critical points.

Differentiability

Differentiability is the property of a function that signifies whether it has a derivative at each point in its domain. If a function is differentiable, it means it has a clear slope—an instantaneous rate of change—at every point.
This concept ensures smoothness in the function’s graph, without abrupt angles or breaks.

Key features:

• Differentiability implies continuity. If a function is differentiable at a point, it is also continuous there, but the reverse isn’t always true.
• When proving limits, such as right-hand and left-hand limits, knowing that a function is differentiable helps reassure that these limits exist.
• If derivatives are bounded, as discussed, differentiability across an interval will often lead to finite limits at the edges of the interval.

Understanding differentiability helps in navigating other complex areas in calculus, providing a firm foundation for solving intricate mathematical problems.