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

If not, provide a counterexample. Answer: Yes, a function that is left-continuous and right-continuous at a point \(a\) is necessarily continuous at point \(a\).

Step-by-step solution

Statement a - Continuity from Left and Right Continuity

Recall that a function \(f\) is continuous at \(a\) if \(\lim_{x \to a} f(x) = f(a)\). A function is left-continuous at \(a\) if \(\lim_{x \to a^-} f(x) = f(a)\), and similarly, right-continuous if \(\lim_{x \to a^+} f(x) = f(a)\). If both left and right limits exist and are equal to \(f(a)\), we have that \(\lim_{x \to a} f(x) = f(a)\), which implies the function is continuous at \(a\). Therefore, statement a is true.

Statement b - Left/Right Continuity from Continuity

We already established that a continuous function implies the existence of both left and right limits. Since the function is continuous, we know that \(\lim_{x \to a} f(x) = f(a)\). This condition implies that the function has a limit from both the left and right sides. Therefore, statement b is also true.

Statement c - Intermediate Value in \((a, b)\)

Statement c is a special case of the Intermediate Value Theorem (IVT), which states that if a continuous function \(f\) is defined on a closed interval \([a, b]\) and \(L\) is any number between \(f(a)\) and \(f(b)\), there must be at least one point \(c\) in \((a, b)\) such that \(f(c) = L\). Since the hypothesis doesn't mention the continuity of the function, the presence of counterexamples is possible. Consider the discontinuous step function \(f(x) = \lfloor x \rfloor,\) where \(a = 1\) and \(b=2\). In this case, \(f(a) = 1 \leq L = 1.5 \leq f(b) = 2\), but there's no point \(c\) in \((a, b)\) for which \(f(c) = L = 1.5\). Therefore, statement c is false.

Statement d - Average Point in a Continuous Function

Statement d is an instance of the Mean Value Theorem (MVT). The MVT states that if a continuous function is defined on a closed interval \([a, b]\) and differentiable on an open interval \((a, b)\), there exists a point \(c\) in \((a, b)\) such that \(f'(c) = \frac{f(b) - f(a)}{b - a}\). While the MVT requires differentiability, since the hypothesis of statement d mentions only continuity, we can consider the continuous but non-differentiable function \(f(x) = |x|\) with \(a = -1\) and \(b = 1\). Here, \(f(a) = f(b) = 1\) and \(f(0) = (f(a) + f(b)) / 2 = 1\). So, there exists a point \(c = 0\) in \((a, b)\) such that \(f(c) = (f(a) + f(b)) / 2.\) Statement d is true for the given function but may not be true for all continuous functions. Therefore, statement d is not generally true. In summary: a. True b. True c. False (counterexample provided) d. Not generally true

Key concepts

Left-Continuity

Understanding the concept of left-continuity is crucial in grasping continuity at a point. A function is considered left-continuous at a point \(a\) if the limit of the function as it approaches \(a\) from the left matches the value of the function at \(a\). Mathematically, this is written as \(\lim_{x \to a^-} f(x) = f(a)\). In simpler terms, if you approach \(a\) from values less than \(a\), the function's output should seamlessly meet the value at \(a\). This indicates that there are no sudden jumps or breaks in the function as you approach \(a\) from the left.

• Check the limit from the left: \(\lim_{x \to a^-} f(x)\)
• Compare it to \(f(a)\): Should be equal for left-continuity.

Acknowledging when a function is left-continuous can help identify overall continuity when combined with right-continuity.

Right-Continuity

Similar to left-continuity, right-continuity ensures that a function behaves predictably from the right side of a point \(a\). A function is right-continuous at \(a\) if \(\lim_{x \to a^+} f(x) = f(a)\). This means as you approach \(a\) from larger values, the function should align perfectly with \(f(a)\). It's like steadily walking towards a doorway from the right and making sure you end up in the room without obstruction once you cross the doorway.

• Right-hand limit: \(\lim_{x \to a^+} f(x)\)
• Must equal \(f(a)\) to hold right-continuity.

Both left and right continuity are necessary conditions for overall continuity at a point \(a\). When combined, they certify continuous behavior.

Intermediate Value Theorem

The Intermediate Value Theorem (IVT) is a pivotal concept in calculus. It asserts that if a function \(f\) is continuous over a closed interval \([a, b]\), then for any value \(L\) between \(f(a)\) and \(f(b)\), there exists a point \(c\) within the interval \(a < c < b\) where \(f(c) = L\). This theorem assures that continuous functions have no gaps or jumps within their intervals.

• Continuity over \([a, b]\) is crucial for the IVT to apply.
• Ensures the value \(L\) is attainable within the interval if it lies between \(f(a)\) and \(f(b)\).

Understanding the Intermediate Value Theorem helps in recognizing the smooth, unbroken character of continuous functions and their ability to take on any intermediate value within a given interval.

Mean Value Theorem

The Mean Value Theorem (MVT) presents a gem in differential calculus. It highlights that for a function that's continuous on a closed interval \([a, b]\) and differentiable on an open interval \((a, b)\), there exists a point \(c\) where the instantaneous rate of change (the derivative) at \(c\) equals the average rate of change over \([a, b]\). Mathematically, \(f'(c) = \frac{f(b) - f(a)}{b - a}\).

• Function must be differentiable in \((a, b)\) and continuous in \([a, b]\).
• Reflects the average slope is realized as the instantaneous slope at some point.

The Mean Value Theorem connects the dots between derivatives and averages, showcasing a function's predictability by guaranteeing this key relationship.