Problem 2 - Calculus: Early Transcendentals

Question

Give the three conditions that must be satisfied by a function to be continuous at a point.

Step-by-Step Solution

Answer

Question: State the three conditions required for a function to be continuous at a particular point. Answer: The three conditions for a function to be continuous at a point x = a are: 1. The limit of the function must exist as x approaches a. 2. The function is defined at the point x = a. 3. The limit of the function as x approaches a equals the function value at a, i.e., \(\lim\limits_{x \to a} f(x) = f(a)\).

Step 1: Condition 1: The Limit of the Function Exists

The first requirement for a function to be continuous at a point is that the limit of the function must exist at that point. In other words, as the input approaches the given point, the function should approach a specific value. Mathematically, we can express this as follows: If the function f is continuous at a point x = a, then the limit as x approaches a of f(x) must exist: \[ \lim\limits_{x \to a} f(x) \]

Step 2: Condition 2: The Function is Defined at the Point

The second requirement for continuity is that the function must be defined at the specific point x = a. This means that the function should have a value at x = a, which we can denote as f(a).

Step 3: Condition 3: The Limit Equals the Function Value

Finally, the third requirement for continuity is that the value of the limit, as x approaches a, must equal the function's value at a: \[ \lim\limits_{x \to a} f(x) = f(a) \] If all these three conditions are satisfied, the function is continuous at the point x = a.

Key Concepts

Limit of a Function

The limit of a function is a fundamental concept in calculus and analysis. When we talk about limits, we're exploring the behavior of a function as the input approaches a certain point. It's like asking, "As I get closer to this point, what value is the function aiming to reach?" For continuity at a point, this limit must exist.In simpler terms, consider a curve on a graph. If, as you move along the curve toward a certain point, the curve approaches a specific height or value, that is the limit. In mathematical language, if you have a function \( f \) and you're interested in the point \( x = a \), the limit of the function as \( x \) approaches \( a \) is expressed as:- \[ \lim\limits_{x \to a} f(x) \]This expression tells us that as the input \( x \) nears the point \( a \), \( f(x) \) should settle on a particular number. If no such number exists, or the values keep bouncing without settling, then the limit does not exist at that point.

Function Defined at a Point

A function must be defined at the point we're interested in if we want to talk about its continuity there. This means that we should be able to plug in the value of \( x = a \) into the function and get an actual number out.For example, if someone hands you a function like \( f(x) = 2x + 3 \) and asks you to evaluate it at \( x = 2 \), you can compute it to get \( f(2) = 7 \). Therefore, \( f \) is defined at \( x = 2 \).In situations where the function isn't defined at a point, like if there's a hole or a break in its graph, continuity cannot even be considered at that point. Think of it like having a road with a gap; if you can't even tread on it, you can't talk about walking smoothly across.

Equal Limit and Function Value

Continuity requires that the limit of the function at the point and the function's actual value at that point must be equal. This equates to a smooth and seamless transition from the values around the point to the value at the point itself.When you compute the limit as \( x \) approaches \( a \) and find it, to confirm continuity,

• Check the actual value of the function at \( x = a \) (let's call it \( f(a) \)).
• Make sure \[ \lim\limits_{x \to a} f(x) = f(a) \]

This matching ensures no unexpected jumps or breaks at \( x = a \). It's the essence of "being continuous," like drawing a line without lifting your pencil off the page. If the limit doesn’t equal the function value, it's like a sudden leap in a road's surface, causing a bump instead of a smooth ride.