Differentiability and continuity

In summary, the conversation discusses how to show that a given function f is differentiable, but its derivative f' is discontinuous at 0. The advice given is to first define f(x) and then solve for f'(x) for both x=0 and x≠0 separately. The Pinching Theorem is suggested as a way to prove that f'(0)=0. It is clarified that f'(0) does exist and the conversation ends with the understanding that the limit being zero does not necessarily mean that the function is continuous at that point.
  • #1
metalbec
6
0
Hi. How do I show that f is differentiable, but f' is discontinuous at 0? I guess I'm just looking for a general idea to show discontinuity.
Thanks
 
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  • #2
metalbec said:
How do I show that f is differentiable, but f' is discontinuous at 0?

First you will have to get clear about what f you have been given. If you are looking at some exercise, it must have given some definition f(x)=...
 
  • #3
well. f(x)= x^2sin(1/x). I can define f(0) to be 0.
 
  • #4
metalbec said:
well. f(x)= x^2sin(1/x). I can define f(0) to be 0.

The derivative becomes a function itself, [itex]f':\mathbb{R}\to\mathbb{R}[/itex]. If you want to prove that it is not continuous, you should first solve its values [itex]f'(x)[/itex] for all x. In this case it is easiest to solve the derivative for [itex]x=0[/itex] and for [itex]x\neq 0[/itex] separately. For [itex]x\neq 0[/itex], you can solve [itex]f'(x)[/itex] by using the usual derivation rules. To solve [itex]f'(0)[/itex] it is best to use the definition of the derivative.
 
  • #5
Would I use the Pinching Theorem to show that because the limit as h approaches 0 from both sides is 0, then the limit of hsin (1/h) is also 0?
 
  • #6
metalbec said:
Would I use the Pinching Theorem to show that because the limit as h approaches 0 from both sides is 0, then the limit of hsin (1/h) is also 0?

yes! This is how you prove [itex]f'(0)=0[/itex].
 
  • #7
Okay, I think I've done that. But I'm having a hard time grasping it. If I have already established that f'(0) does not exist, how did I just show that it is zero? Or does the fact that its limit is zero in itself show that it is discontinuous at 0?
 
  • #8
metalbec said:
If I have already established that f'(0) does not exist

You have done a mistake here. f'(0) exists.
 

1. What is the difference between differentiability and continuity?

Differentiability is a property of a function where it has a well-defined derivative at a given point. Continuity, on the other hand, refers to a function's ability to have a smooth and unbroken graph without any abrupt changes. In simpler terms, differentiability deals with the slope of a function, while continuity deals with the connectedness of its graph.

2. How can I determine if a function is differentiable at a given point?

A function is differentiable at a point if the limit of its derivative exists at that point. This means that the right-hand and left-hand derivatives must be equal at that point, and the function must be continuous at that point as well.

3. Can a function be continuous but not differentiable?

Yes, a function can be continuous but not differentiable at a given point. This happens when the function has a sharp turn or corner at that point, making the derivative undefined. A classic example of this is the absolute value function, which is continuous but not differentiable at x=0.

4. What is the importance of differentiability and continuity in mathematics and real-life applications?

Differentiability and continuity are essential concepts in calculus and are used to analyze and model real-world phenomena in fields such as physics, engineering, and economics. They allow us to understand and predict the behavior of functions and their rates of change, which are crucial in various applications, such as optimization and motion analysis.

5. Are there any functions that are neither differentiable nor continuous?

Yes, there are functions that are neither differentiable nor continuous. These are called "discontinuous" functions and can have various forms, such as jump discontinuities, removable discontinuities, and oscillating discontinuities. A classic example is the Dirichlet function, which is continuous at irrational numbers but discontinuous at rational numbers.

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