Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 2: Problem 29
Determine whether the statement is true or false. Explain your answer. If a function \(f\) is continuous at \(x=0,\) then \(f\) is differentiable at \(x=0\)
Short Answer
Expert verified
False: A function can be continuous at a point without being differentiable, as shown by the example of \( f(x) = |x| \).
Step by step solution
01
Define Continuity
A function \( f \) is continuous at a point \( x = 0 \) if the following condition is satisfied: \( \lim_{x \to 0} f(x) = f(0) \). This means the limit of \( f \) as \( x \) approaches 0 equals the value of the function at 0.
02
Define Differentiability
A function \( f \) is differentiable at a point \( x = 0 \) if the derivative \( f'(0) \) exists. This is defined as: \( f'(0) = \lim_{h \to 0} \frac{f(0+h) - f(0)}{h} \).
03
Understand the Implication
The statement claims that if a function is continuous at \( x=0 \), it must also be differentiable there. In essence, it suggests continuity implies differentiability.
04
Provide a Counterexample
Consider the function \( f(x) = |x| \). This function is continuous at \( x=0 \), since \( \lim_{x \to 0} |x| = 0 = |0| \). However, \( f(x) \) is not differentiable at \( x=0 \) because the derivative from the left is \( -1 \) and from the right is \( 1 \), causing a sharp corner at \( x=0 \). Since the left-hand and right-hand limits are not equal, \( f'(0) \) does not exist.
05
Conclude False Statement
Since a function like \( f(x) = |x| \) is continuous at \( x=0 \) but not differentiable there, the statement "if a function \( f \) is continuous at \( x=0 \), then \( f \) is differentiable at \( x=0 \)" is false.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Continuity
Continuity is a fundamental concept in calculus that describes how smooth a function behaves around a certain point. For a function \( f \) to be continuous at a specific point \( x = 0 \), there is a key condition that must be satisfied: the limit of the function as \( x \) approaches 0 must equal the function鈥檚 value at that point. Mathematically, this is expressed as \( \lim_{x \to 0} f(x) = f(0) \).
- Essentially, if you move towards \( x = 0 \) from either direction, the values of \( f(x) \) get arbitrarily close to \( f(0) \).
- This means there shouldn't be any jumps or gaps in the function's graph at \( x = 0 \).
Differentiability
Differentiability is a concept that builds on continuity and takes it a step further. When we say a function \( f \) is differentiable at a point \( x = 0 \), it means that the function has a defined tangent at that point, which mathematically translates into the existence of a derivative. The derivative at this point is given by\[f'(0) = \lim_{h \to 0} \frac{f(0+h) - f(0)}{h}\]
- This expression essentially measures how \( f(x) \) changes as \( x \) changes, giving an instantaneous rate of change or slope of the function at \( x = 0 \).
- If this limit exists, \( f \) is differentiable at that point, implying the function doesn鈥檛 have any sharp corners or cusps at \( x = 0 \).
Counterexample
To truly understand why not all continuous functions are differentiable, we can look at counterexamples. A counterexample provides evidence that a general statement is false. Consider the function \( f(x) = |x| \) at \( x = 0 \).
- \( f(x) = |x| \) is continuous at \( x = 0 \) because \( \lim_{x \to 0} |x| = 0 = |0| \), showing that there is no jump or break in the graph of \( f \) at this point.
- However, \( f(x) = |x| \) is not differentiable at \( x=0 \), due to the behavior of its derivative.
- The derivative from the left approaches \(-1\), and from the right, it approaches \(1\). These differing values indicate a cusp, illustrating that the slope changes abruptly at \( x = 0 \), and thus \( f'(0) \) does not exist.
Limit
The concept of a limit is vital in discussing both continuity and differentiability. In calculus, a limit describes the value that a function \( f(x) \) approaches as the input \( x \) gets arbitrarily close to a specific point.
When we say \( \lim_{x \to 0} f(x) \), we are interested in what happens to \( f(x) \) as \( x \) nears 0. This principle is foundational for defining continuity, as a function is continuous at a point if its limit at that point equals the function's actual value.
When we say \( \lim_{x \to 0} f(x) \), we are interested in what happens to \( f(x) \) as \( x \) nears 0. This principle is foundational for defining continuity, as a function is continuous at a point if its limit at that point equals the function's actual value.
- Limits also play an essential role in defining differentiability, where the limit is used to define the derivative of the function.
- Understanding limits helps us predict and analyze the behavior of functions near specific points, especially where intuitive inspection might fail.
