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Chapter 2: Problem 16

True or False: If a function is continuous at a number, then it is differentiable at that number.

Short Answer

Expert verified
False. Continuity does not imply differentiability.

Step by step solution

01

Understanding Continuity

A function is continuous at a number, say \( x = a \), if the limit of the function as \( x \) approaches \( a \) is equal to the value of the function at \( a \). Formally, a function \( f \) is continuous at \( x = a \) if \( \lim_{x \to a} f(x) = f(a) \). This means there are no breaks, jumps, or holes at \( a \).
02

Understanding Differentiability

A function is differentiable at a point \( x = a \) if the derivative \( f'(a) \) exists at that point. This requires that the function is smooth and not just continuous; it implies the function has a well-defined tangent at that point without any sharp corners or cusps.
03

Relationship Between Continuity and Differentiability

While differentiability implies continuity (if a function is differentiable at \( a \), then it must be continuous at \( a \)), the converse is not always true. Continuity does not guarantee differentiability. A classic counterexample is the absolute value function \( f(x) = |x| \), which is continuous at \( x = 0 \), but not differentiable there.
04

Conclusion

Based on the analysis, the statement "If a function is continuous at a number, then it is differentiable at that number" is false. There exist functions which are continuous at a point but not differentiable at the same point.

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Key Concepts

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

Differentiability
Knowing if a function is differentiable at a point is a key aspect of calculus. Differentiability at a point means the function has a well-defined derivative at that specific location. In simpler terms, the function should be 'smooth' at that point. No sharp turns or corners. Think of the slope of a curve; differentiability ensures the slope (or rate of change) exists.
  • A function is differentiable at a number, say \( x = a \), if \( f'(a) \) exists.
  • This implies the function needs to be continuous at \( a \), but more importantly, it can't have any abrupt changes.
  • If a function isn't smooth, it means it could have sharp edges, like the vertex of the absolute value function \( f(x) = |x| \).
To sum up, for a function to be differentiable at a point, it must be "smooth", not just connected or complete at that point.
Continuity
Continuity is one of the foundational concepts in calculus, describing a function that is unbroken or uninterrupted at a certain point. If a function is continuous at \( x = a \), it means the graph of the function travels smoothly through \( a \) without any breaks or jumps.
  • For continuity at a specific point \( x = a \), three conditions must be true:
  • First, \( f(a) \) must be defined. The function needs to have a real output at \( a \).
  • Second, \( \lim_{x \to a} f(x) \) must exist. As \( x \) approaches \( a \), \( f(x) \) approaches the same limit.
  • Finally, \( \lim_{x \to a} f(x) = f(a) \). The limit as \( x \) approaches \( a \) should match the function's actual value at \( a \).
Understanding these criteria helps ensure the function doesn't have gaps or sudden jumps at any point.
Limits
Limits are essential to understanding both continuity and differentiability. A limit refers to the value that a function approaches as the input approaches some value. They are key to solving calculus problems related to continuity and differentiability.
  • The notation \( \lim_{x \to a} f(x) = L \) means that as \( x \) approaches \( a \), the function \( f(x) \) gets closer to \( L \).
  • Limits help determine the behavior of functions near points of interest, especially where the function might not be explicitly defined.
  • If the right-hand and left-hand limits at a point converge to the same number, the limit at that point exists.
When you understand limits, you have a solid groundwork for tackling advanced calculus topics.

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Most popular questions from this chapter

Using your own words, explain geometrically why the derivative is undefined where a curve has a vertical tangent.

Use the Generalized Power Rule to find the derivative of each function. $$f(x)=x^{2} \sqrt{1+x^{2}}$$

Graph the cost function \(y_{1}=\sqrt{4 x^{2}+900}\) on the window \([0,30]\) by \([-10,70]\). Then use NDERIV to define \(y_{2}\) as the derivative of \(y_{1}\). Verify the answer to Exercise 53 by evaluating the marginal cost function \(y_{2}\) at \(x=20\).

True or False: If a function is differentiable at a number, then it is continuous at that number.

Find the derivative of \(\left(x^{2}+1\right)^{2}\) in two ways: a. By the Generalized Power Rule. b. By "squaring out" the original expression and then differentiating. Your answers should agree.
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