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Magazine Chapter 1: Problem 81
Show \(f(x)=x\) is continuous everywhere.
Short Answer
Expert verified
The function is continuous everywhere because its limit equals its value at any point.
Step by step solution
01
Definition of Continuity
A function is continuous at a point \(x = c\) if the following condition is met: \[ \lim_{{x \to c}} f(x) = f(c). \] That means as \(x\) approaches \(c\), the function approaches \(f(c)\). We need to show this condition satisfied for every \(c\) in the domain of \(f(x) = x\).
02
Evaluate the Function at the Point c
For the function \(f(x) = x\), the value at a point \(c\) is simply \(f(c) = c\). This will serve as the right side of our continuity condition.
03
Find the Limit as x Approaches c
Evaluate the limit \( \lim_{{x \to c}} f(x) \). For \(f(x) = x\), this becomes \( \lim_{{x \to c}} x = c\), by basic properties of limits since the function \(x\) is linear.
04
Verify the Continuity Condition
Confirm the two sides of the continuity condition are equal: \( \lim_{{x \to c}} f(x) = c = f(c) \). Since these are identical expressions, the function meets the criteria for continuity at \(x = c\).
05
Conclude Continuity for All x
Since \(c\) was any arbitrary point in the domain, and since we verified the continuity condition holds for this arbitrary point, we can conclude that \(f(x) = x\) is continuous at every point in its domain, which is all real numbers.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Function Limit
The concept of a function limit is essential in understanding continuity. When we talk about the limit of a function, we want to see how the function behaves as the input value approaches a specific point. This is usually denoted as \( \lim_{{x \to c}} f(x) \), which means we are looking at the value that \( f(x) \) gets closer to as \( x \) gets closer to \( c \). For example, with the function \( f(x) = x \), we want to find what happens to \( f(x) \) when \( x \) approaches a point \( c \). Since \( f(x) = x \), when \( x \) approaches \( c \), \( f(x) \) simply approaches \( c \).
- The idea of a limit helps us predict the behavior of a function close to a specific point without having to substitute that point directly.
- It is a key building block for defining what it means for a function to be continuous.
Point Evaluation
Point evaluation in the context of continuity is straightforward. It involves determining what a function equals at a particular point. Let's say we have a function \( f(x) \) and we evaluate it at a point \( c \). This would mean calculating \( f(c) \). For our function \( f(x) = x \), point evaluation is particularly simple. Wherever we choose a point \( c \), the function value at this point is \( f(c) = c \). This is because the identity function \( f(x) = x \) has the same input and output. Point evaluation allows us to find the exact output of a function at specific points:
- This helps in checking if the limit of the function as \( x \) approaches \( c \) is actually equal to \( f(c) \).
- Therefore, it is a crucial step in verifying continuity at that point.
Real Numbers
Real numbers are the backbone of many mathematical concepts, including continuity. They encompass both rational numbers (like 3 or 3/4) and irrational numbers (like \( \sqrt{2} \) or \( \pi \)). The domain of the function \( f(x) = x \) includes all real numbers. That means any real number can be an input to this function, making it infinitely continuous. Real numbers feature some important properties that are useful:
- They are dense, meaning between any two real numbers, there is another real number. This fact plays a role in the idea of limits.
- They provide a complete field, useful for smooth transitions around any selected point \( c \).
Arbitrary Point
The idea of an arbitrary point is very important when discussing continuity. It involves selecting a generic point in the domain of a function to test whether a given property holds true. When we say the function \( f(x) = x \) is continuous everywhere, we mean it is continuous at every arbitrary point \( c \) on the real number line.This doesn't need a specific number to be plugged in. Instead, we rely on the properties of the function itself at any such point.
- This method guarantees the function is continuous throughout its domain as we don鈥檛 rely on a special case or exception.
- It provides a form of generality proving the function behaves the same way across all inputs.
