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Chapter 4: Problem 2

(a) Prove that the function \(f(x)=1, x \in \mathbb{R}\), is continuous on \(\mathbb{R}\). (b) Prove that the function \(f(x)=x, x \in \mathbb{R}\), is continuous on \(\mathbb{R}\).

Short Answer

Expert verified
(a) The function \( f(x) = 1 \) is continuous on \( \mathbb{R} \). (b) The function \( f(x) = x \) is continuous on \( \mathbb{R} \).

Step by step solution

01

Understanding the Function

We have the function \( f(x) = 1 \) for which we need to prove continuity. A function is continuous if the limit as \( x \to c \) is equal to the function value at \( c \), that is \( \lim_{x \to c} f(x) = f(c) \) for all \( c \in \mathbb{R} \).
02

Evaluating the Limit

For any \( c \in \mathbb{R} \), we have \( \lim_{x \to c} f(x) = \lim_{x \to c} 1 = 1 \). Since the limit is a constant function, it simplifies to 1 directly. Thus, \( \lim_{x \to c} f(x) = 1 \).
03

Checking Continuity at Any Point

The function value at any point \( c \) is \( f(c) = 1 \). For the function to be continuous at \( c \), \( \lim_{x \to c} f(x) = 1 \) should equal \( f(c) = 1 \). Hence, \( f(x) \) is continuous at all \( c \in \mathbb{R} \).
04

Understanding the Linear Function

Now we consider the function \( f(x) = x \). Again, for continuity, we check if \( \lim_{x \to c} f(x) = f(c) \) for all \( c \in \mathbb{R} \).
05

Calculating the Limit

For \( f(x) = x \), the limit as \( x \to c \) is \( \lim_{x \to c} x = c \). This is because replacing \( x \) with \( c \) directly gives the value \( c \).
06

Verifying Continuity

The function value at any point \( c \) is \( f(c) = c \). To show continuity, we need \( \lim_{x \to c} x = c = f(c) \). Therefore, \( f(x) = x \) is continuous at every \( c \in \mathbb{R} \).

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Key Concepts

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

constant function
A constant function is one of the simplest types of functions in mathematics. It is defined as a function where the output value is the same regardless of the input value. For example, the function \( f(x) = 1 \) is a constant function because, for every real number \( x \), the output is always 1. Constant functions are quite straightforward because:
  • They do not change as the input varies.
  • Their graphs are horizontal lines on the coordinate plane.
  • They are defined for all real numbers, thus \( x \) can be any real number.
This simplicity makes constant functions always continuous, as the value never changes, and hence the limit at any point \( c \) always equals the function value at \( c \). Thus, \( \lim_{x \to c} f(x) = f(c) = 1 \) for our example.
limit of a function
The limit of a function is a fundamental concept in calculus. It describes the behavior of a function as the input approaches a particular value. In simpler terms, it tells us what value a function gets closer to as the input approaches some point \( c \).
  • If \( \lim_{x \to c} f(x) \) exists and equals \( L \), it means as \( x \) approaches \( c \), \( f(x) \) gets arbitrarily close to \( L \).
  • For constant functions like \( f(x) = 1 \), the limit as \( x \) approaches any real number \( c \) is simply the constant itself, which is 1.
  • The calculation simplifies because the value never changes; it鈥檚 always the same, making it easy to determine the limit.
Limits are crucial for defining the continuity of functions and analyzing their behaviors as inputs change.
continuous function on real numbers
A continuous function on real numbers means the function does not have any sudden jumps or breaks at any point within the domain. For a function \( f(x) \) to be continuous at a point \( c \), the following conditions need to be satisfied:
  • \( \lim_{x \to c} f(x) \) must exist.
  • \( f(c) \) must be defined.
  • The limit of the function as \( x \) approaches \( c \) must equal the function value at \( c \), that is \( \lim_{x \to c} f(x) = f(c) \).
For example, \( f(x) = x \) is continuous on the set of real numbers \( \mathbb{R} \) because:
  • For any real number \( c \), both the left-hand and right-hand limits exist and are equal to \( c \).
  • \( f(c) \) is also \( c \), so the limit equals the function value.
Continuous functions are pivotal in calculus because they ensure smoothness and predictability across their domain.
proof of continuity
Proving continuity involves demonstrating that the conditions for continuity hold for all points in the domain of the function. Here's a step-by-step approach used in mathematical proofs:
  • Identify the function you want to prove is continuous. For example, proving \( f(x) = 1 \) is continuous involves showing it has no breaks or jumps for any \( x \) in \( \mathbb{R} \).
  • Calculate \( \lim_{x \to c} f(x) \) for any \( c \), ensuring this limit exists. For a constant function, the limit is simply the constant value, such as 1 for \( f(x) = 1 \).
  • Verify that \( f(c) \) is defined and equals the limit for continuity. Since \( f(c) = 1 \) and \( \lim_{x \to c} f(x) = 1 \), continuity holds across all \( c \).
  • Apply these principles to other functions, such as \( f(x) = x \), ensuring the functions satisfy continuity鈥檚 requirement throughout their domain.
Approaching continuity proofs systematically ensures a solid understanding and allows for extension to more complex functions.

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Most popular questions from this chapter

Prove that the function \(f(x)=\sin \left(\frac{\pi}{2} \cos x\right)\) is continuous on \(\mathbb{R}\).

Prove that the following function is continuous on \(\mathbb{R}\), stating each rule or fact about continuity that you are using $$ f(x)=\cos \left(x^{5}-5 x^{2}\right)+7 e^{-x^{2}} $$

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(a) Determine whether the function \(f(x)=x^{3}-2 x^{2}, x \in \mathbb{R}\), is continuous at 2 . (b) Determine whether the function \(f(x)=[x], x \in \mathbb{R}\), is continuous at 1 .
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