Question

Determine whether the following statements are true and give an explanation or counterexample. a. If a function is left-continuous and right-continuous at \(a,\) then it is continuous at \(a\). b. If a function is continuous at \(a,\) then it is left-continuous and right- continuous at \(a\). c. If \(a

Short answer

Question: Determine the truth value of the following statement and provide an explanation or counterexample: If a function is left-continuous and right-continuous at a point, then it is continuous at that point. Answer: True. If a function is left-continuous and right-continuous at a point, the limits from both the left and right side exist and are equal to the function value at that point, making the function continuous at that point.

Step-by-step solution

Statement a. If a function is left-continuous and right-continuous at \(a,\) then it is continuous at \(a\).

This statement is true. According to the definitions of left-continuity and right-continuity, a function is left-continuous at \(a\) if the limit as \(x\) approaches \(a\) from the left exists and is equal to the function value at \(a\), and right-continuous at \(a\) if the limit as \(x\) approaches \(a\) from the right exists and is equal to the function value at \(a\). If a function is left-continuous and right-continuous at \(a,\) then the limits from both the left and right side exist and are equal to the function value at \(a\), so the overall limit as \(x\) approaches \(a\) exists and is equal to the function value at \(a\), making the function continuous at \(a.\)

Statement b. If a function is continuous at \(a,\) then it is left-continuous and right-continuous at \(a\).

This statement is true. If a function is continuous at \(a\), then the limit as \(x\) approaches \(a\) exists and is equal to the function value at \(a\). Since the overall limit exists, it necessarily means that both the left and right limits must also exist and equal the function value at \(a.\) Therefore, the function is left-continuous and right-continuous at \(a.\)

Statement c. If \(a

This statement is false. The given information is not sufficient to guarantee that there will be some value \(c\) in \((a, b)\) for which \(f(c) = L\). A counterexample is the function \(f(x) = x^2\) on the interval \([0,2]\). We have that \(f(0) = 0\), \(f(2) = 4\), and \(L=3\). There is no value of \(c\) in \((0,2)\) for which \(f(c) = 3\) since \(f(x) = x^2\) is an increasing function on that interval.

Statement d. Suppose \(f\) is continuous on [a, b]. Then there is a point \(c\) in \((a, b)\) such that \(f(c)=(f(a)+f(b)) / 2\).

This statement is true, assuming \(f(a) \neq f(b)\). By the Intermediate Value Theorem, when a function is continuous on a closed interval \([a,b]\), it takes on all values between \(f(a)\) and \(f(b)\) over that interval. Specifically, when \(f(a) \neq f(b)\), we know that there exists a point \(c\) in the interval \((a, b)\) where the function value is halfway between \(f(a)\) and \(f(b)\). Thus, there exists a point \(c\) where \(f(c)=(f(a)+f(b)) / 2\).

Key concepts

Left-Continuity

In mathematics, the concept of left-continuity at a point is an essential building block in understanding how functions behave. We say a function is left-continuous at a point \(a\) if the limit of the function as \(x\) approaches \(a\) from the left (denoted as \(x \to a^-\)) exists and equals the function's value at \(a\), written as \( \lim_{{x \to a^-}} f(x) = f(a) \).
This means if we move towards \(a\) from values less than \(a\), the function's value approaches exactly \(f(a)\).

Why is this important? Left-continuity helps ensure that there are no sudden jumps or breaks in the function as we approach from the left side. This means the function is predictable in that direction, which is useful in mathematical analysis and applications. In real-world applications, such behaviors show stability in systems represented by the function.

In summary:

• Definition: Left-continuous at \(a\) if \(\lim_{{x \to a^-}} f(x) = f(a)\).
• Implication: No sudden jumps or breaks from the left side.
• Usage: Ensures stable predictions approaching from the left.

Right-Continuity

Right-continuity is the sibling of left-continuity and just as important in understanding the behavior of functions. A function is said to be right-continuous at a point \(a\) if the limit of the function as \(x\) approaches \(a\) from the right (denoted by \(x \to a^+\)) exists and equals the function's value at \(a\), which is expressed as \( \lim_{{x \to a^+}} f(x) = f(a) \).
This ensures that the function smoothly reaches the value \(f(a)\) without any interruptions or abrupt changes when approached from values greater than \(a\).

Right-continuity is crucial in maintaining consistent predictions and calculations when approaching from the right side of a function. Much like left-continuity, it is an integral part of making sure a function is fully continuous at a point. When both left and right continuity are satisfied at \(a\), the function is simply continuous at \(a\).

To recap:

• Definition: Right-continuous at \(a\) if \(\lim_{{x \to a^+}} f(x) = f(a)\).
• Implication: Smooth and predictable behavior from the right side.
• Integration: Part of overall continuity along with left-continuity.

Intermediate Value Theorem

The Intermediate Value Theorem (IVT) is a powerful concept in calculus that relates to how continuous functions behave over an interval.
The theorem states that if a function \(f\) is continuous on a closed interval \([a, b]\), and \(N\) is any number between \(f(a)\) and \(f(b)\), then there exists at least one point \(c\) in the open interval \((a, b)\) such that \(f(c) = N\).
This can seem like an abstract concept, but essentially, it guarantees that continuous functions will hit every height between the values at the start and the end of the interval.

Why does IVT matter? It ensures that continuous functions map out a "bridge" over the interval, leaving no gaps in the possible values they can achieve. This property has numerous applications from proving the existence of solutions to equations to ensuring smoother transitions in engineering designs.

To summarize:

• Core Idea: A function covers all intermediate values between any two points where it is continuous.
• Practicality: Guarantees solutions and consistency across intervals.
• Applications: Used in proofs, solutions, and design considerations.