Proof of Smoothness: Analytical Steps & Examples

In summary, the conversation is about proving that a specific function, G(x), is smooth. The method for proving this is by using rules such as sums, products, and compositions of smooth functions, or by using induction on the derivatives. The specific function being discussed is G(x) = f'(0) when x=0 and (f(x)-f(0))/x when x is not equal to zero. The question is how to prove that this function is smooth at x=0.
  • #1
nughret
45
0
My lecturer gave me a question that included giving a proof that a particular function is smooth. I have taken a course on analysis and have no problems when it comes to proof of continuity; i was just wondering what the usual steps are in proving that a function is smooth.
I would guess that it would involve some sort of induction on the derivatives but if someone could sketch out a general proof that would be grand.
 
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  • #2
There are some rules like: sums, products and compositions of smooth functions are smooth. Usually one uses these to show that a function is smooth. Otherwise, induction on the derivatives might work.

Since it is only a small portion of your entire exercise, could you post the function?
 
  • #3
Yes well this specific question is stated as such :
f(x) is a smooth function, prove the function

G(x) = f'(0) , x = 0
(f(x) - f(0))/x , otherwise

is smooth.
I previously assumed by product rule it is true that G(x) is smooth when x is not equal to zero but obviously the whole point of the question is about x=0.
 

1. What is "proof of smoothness" and why is it important in science?

"Proof of smoothness" refers to the process of mathematically proving that a function is continuous and differentiable at every point within its domain. It is important in science because it allows us to confidently use these functions in calculations and models, knowing that they will behave predictably and accurately.

2. How do you analytically prove the smoothness of a function?

To analytically prove the smoothness of a function, you must show that it is both continuous and differentiable at every point in its domain. This can be done by evaluating the limit of the function as it approaches a point and showing that it exists, as well as calculating the derivative of the function at that point and showing that it is well-defined.

3. Can you provide an example of a function that is not smooth?

Yes, a function that is not smooth would be one that is either discontinuous or non-differentiable at one or more points in its domain. One example is the absolute value function, which is not differentiable at x=0.

4. What are the implications of a function not being smooth?

If a function is not smooth, it means that it does not have a continuous and well-defined rate of change at one or more points in its domain. This can make it difficult to use in calculations and models, as its behavior may be unpredictable and inaccurate.

5. How is proof of smoothness used in real-world applications?

Proof of smoothness is used in many real-world applications, particularly in fields such as physics, engineering, and economics. It allows us to confidently use mathematical models to make predictions and decisions, knowing that the functions used in these models behave smoothly and accurately reflect the real-world phenomena they are trying to describe.

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