Question

Complete the following sentences in terms of a limit. a. A function is continuous from the left at \(a\) if ________. b. A function is continuous from the right at \(a\) if ________.

Short answer

Question: Define left continuity and right continuity in terms of limits, and complete the following sentences: a. A function is continuous from the left at \(a\) if ______________. b. A function is continuous from the right at \(a\) if ______________. Answer: a. A function is continuous from the left at \(a\) if \(\lim_{x \to a^-} f(x) = f(a)\). b. A function is continuous from the right at \(a\) if \(\lim_{x \to a^+} f(x) = f(a)\).

Step-by-step solution

(Step 1: Define left continuity.)

(Left continuity is a property of a function at a certain point, where the function approaches the same value when the argument approaches the point from the left as the value of the function at that point. In terms of a limit, we should express this definition as: A function \(f(x)\) is continuous from the left at \(a\) if \(\lim_{x \to a^-} f(x) = f(a)\). This means that the limit of the function as x approaches a from the left (denoted by the negative superscript) is equal to the function's value at the point a.)

(Step 2: Define right continuity.)

(Right continuity is another property of a function at a certain point, where the function approaches the same value when the argument approaches the point from the right as the value of the function at that point. Similar to left continuity in terms of a limit, we can express this definition as: A function is continuous from the right at \(a\) if \(\lim_{x \to a^+} f(x) = f(a)\). This means that the limit of the function as x approaches a from the right (denoted by the positive superscript) is equal to the function's value at the point a.) So, the completed sentences are: a. A function is continuous from the left at \(a\) if \(\lim_{x \to a^-} f(x) = f(a)\). b. A function is continuous from the right at \(a\) if \(\lim_{x \to a^+} f(x) = f(a)\).

Key concepts

Left Continuity

Left continuity refers to the behavior of a function as it approaches a particular point from the left side. Imagine walking towards a goalpost from the left—left continuity ensures that, as you reach the post, you end up exactly where you intended. For a function to be left continuous at a point \(a\), the following must hold true:

• The left-hand limit exists as \(x\) approaches \(a\) from values less than \(a\).
• The left-hand limit is equal to the actual value of the function at that point \(f(a)\).

In simpler terms, as \(x\) moves closer to \(a\) from the left, \(f(x)\) should smoothly transition into \(f(a)\), ensuring no jumps or discontinuities. Thus, for left continuity, we say \(\lim_{x \to a^-} f(x) = f(a)\). This concept helps us analyze functions' behaviors in one direction, providing insights into their overall continuity.

Right Continuity

Right continuity examines a function's behavior as it approaches a particular point from the right. Think about arriving at the same goalpost, but this time from the right side. For the function to be right continuous at point \(a\), the following conditions should be met:

• The right-hand limit exists as \(x\) approaches \(a\) from values greater than \(a\).
• This limit is exactly equal to the value of the function at that point \(f(a)\).

This means that as \(x\) gets closer to \(a\) coming from the right, \(f(x)\) should seamlessly become \(f(a)\). It ensures there are no surprises or foreseen jumps when transitioning from one side to the other at point \(a\). Therefore, one can express right continuity as \(\lim_{x \to a^+} f(x) = f(a)\). This concept is pivotal in understanding continuous flows from one side of the parameter space.

Limits

Limits are an essential concept in calculus used to describe how a function behaves as its input approaches a certain value. They help determine precisely whether a function has any gaps, jumps, or breaks at any given point, thus establishing continuity. We use limits to define both left and right continuity:

• For left continuity, it involves \(\lim_{x \to a^-} f(x)\).
• For right continuity, it relates to \(\lim_{x \to a^+} f(x)\).

In essence, understanding limits is about predicting what happens to \(f(x)\) as \(x\) endlessly inches closer to \(a\). One key to mastering limits is grasping how they work from each direction, offering insight into the function's approach to a particular value without even needing to get there exactly. Limits provide the fundamental tools needed to dissect and comprehend the delicate nature of continuous functions thoroughly.