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Chapter 12: Problem 113

State in your own words what it means for a function \(f\) to be continuous at \(x=c\)

Short Answer

Expert verified
A function is continuous at \(x=c\) if \(\lim_{{x \to c}} f(x) = f(c)\), meaning no breaks or jumps in the graph.

Step by step solution

01

Define Continuity of a Function

To begin understanding what it means for a function \(f\) to be continuous at \(x=c\), we need to have a clear definition. A function is considered continuous at a point \(x=c\) if there are no breaks, jumps, or holes at that point in its graph. This means the function's value changes smoothly and predictably around \(x=c\).
02

Mathematical Condition for Continuity

In mathematical terms, a function \(f\) is continuous at \(x=c\) if it meets the following condition: \[ \lim_{{x \to c}} f(x) = f(c) \]This means that as \(x\) approaches \(c\) from both sides, the value of \(f(x)\) approaches \(f(c)\). Thus, there is no abrupt change in the value of the function at that point.
03

Understanding the Limit Condition

To ensure continuity at \(x=c\), three things must happen: 1. \(f(c)\) must be defined, meaning \(f\) must have a value at \(c\).2. The limit \(\lim_{{x \to c}} f(x)\) must exist.3. The limit \(\lim_{{x \to c}} f(x)\) must equal \(f(c)\). If any of these conditions do not hold, the function is not continuous at that point.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Limits
Limits form the foundation for understanding continuity. A limit describes the value that a function approaches as the input approaches a certain point. Think of it like this: as you get closer and closer to a particular point on the graph of a function, the limit tells you where that function is headed.
Here's a simple way to picture it:
  • Imagine walking along a path that goes over a hill. The limit tells you how steep that hill gets as you approach the top.
  • If you're approaching a house and you can see it getting closer without any obstacles, you're getting a sense of the limit.
The formal notation is: \[\lim_{{x \to c}} f(x)\]This notation reads as "the limit of \(f(x)\) as \(x\) approaches \(c\)." It helps us determine if a function behaves predictably at a certain point. Without limits, we'd struggle to understand how functions behave when we zoom in on specific points.
Function Definition
Before diving into continuity, it's crucial to know what a function is. A function is like a special relationship between inputs and outputs, where each input has a single output. Think of it as a machine that takes something in and spits something out.
Let鈥檚 break it down:
  • Every function must assign exactly one output for each input.
  • Functions can be represented using graphs, equations, or tables.
For example, if you input 3 into the function \(f\) and the output is 9, it means \(f(3) = 9\). This clear pairing makes functions predictable and essential for deeper mathematical understanding. Understanding this definition helps us identify when a function is well-defined and ready to be analyzed for continuity at a point.
Mathematical Conditions for Continuity
For a function to be continuous at a point \(x = c\), we need to ensure it doesn鈥檛 have any sudden changes. To achieve this, we must meet three essential conditions:
  • Defined Value: The function must have a value at \(x = c\), meaning \(f(c)\) exists.
  • Existing Limit: The limit \(\lim_{{x \to c}} f(x)\) must exist. This indicates predictability and a lack of sudden jumps as \(x\) approaches \(c\).
  • Limit Equals Function Value: The value of the limit must equal the function's value at that point, ensuring smoothness. Therefore, \(\lim_{{x \to c}} f(x) = f(c)\).
If all of these are true, the function is continuous at \(x = c\).
On the other hand, if any are missing, the function isn鈥檛 continuous there, meaning there could be a gap, jump, or hole at \(x = c\). Understanding these conditions helps you spot continuity in functions effectively.

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Most popular questions from this chapter

For the following exercises, evaluate each limit using algebraic techniques. $$\lim _{h \rightarrow 0}\left(\frac{\sqrt{h^{2}+25}-5}{h^{2}}\right)$$

For the following exercises, explain the notation in words when the height of a projectile in feet, \(s\), is a function of time \(t\) in seconds after launch and is given by the function \(s(t)\). $$s(2)$$

For the following exercises, use numerical evidence to determine whether the limit exists at \(x=a\). If not, describe the behavior of the graph of the function at \(x=a\). $$f(x)=\frac{-2}{x-4} ; a=4$$

For the following exercises, evaluate the limits using algebraic techniques. $$\lim _{x \rightarrow-3}\left(\frac{\frac{1}{3}+\frac{1}{x}}{3+x}\right)$$

For the following exercises, evaluate the following limits. $$ \lim _{x \rightarrow 4} \frac{\sqrt{x+5}-3}{x-4} $$
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