Differentiability of composite functions

In summary, the chain rule states that a composition of differentiable functions is differentiable. However, it is not necessarily true that if a composition of functions is differentiable, then all the functions in the composition must be differentiable. Examples can be constructed where one of the functions may be discontinuous or arbitrarily bad, yet the composition remains differentiable. Similarly, a composition may be differentiable even if one of the functions is not. Careful consideration of the individual functions is necessary in determining differentiability of a composition.
  • #1
raphile
23
0
Hi, I have a small question about this. Using the chain rule, I know that a composition of differentiable functions is differentiable. But is it also true that if a composition of functions is differentiable, then all the functions in the composition must be differentiable?

For example, if [itex]f(g(x))[/itex] is differentiable, does that imply [itex]f(x)[/itex] and [itex]g(x)[/itex] are both differentiable?

Thanks!
 
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  • #2
raphile said:
Hi, I have a small question about this. Using the chain rule, I know that a composition of differentiable functions is differentiable. But is it also true that if a composition of functions is differentiable, then all the functions in the composition must be differentiable?

For example, if [itex]f(g(x))[/itex] is differentiable, does that imply [itex]f(x)[/itex] and [itex]g(x)[/itex] are both differentiable?

Thanks!


[itex]\cos\sqrt{x}[/itex] is differentiable from the right at [itex]x=0[/itex] , but [itex]\sqrt{x}[/itex] isn't...

DonAntonio
 
  • #3
Examples can be constructed where f(x) or g(x) can be arbitrarily bad (for example discontinuous) yet f(g(x)) is differentiable

A classic example of this is when f(x)=1 and g(x) is any function you pick. f(g(x))=1 so this composition is differentiable, but g(x) clearly doesn't have to be.

For the other way around consider f(x)="any function which is always negative" if x<1 (for example -x6), and x4 if x >=1. Now let g(x) = x2+3. Because g(x)>=1 always, f(g(x)) is always differentiable (because we are always using the x4 portion of f(x) thanks to how g was constructed), even though a large chunk of f(x) is not differentiable.
 
  • #4
Thanks for the help and examples!
 
  • #5


Hi there,

Thank you for your question. The answer is yes, if a composition of functions is differentiable, then all the functions in the composition must be differentiable. This is because the chain rule states that the derivative of a composition of functions is equal to the product of the derivatives of each individual function in the composition. Therefore, for the derivative to exist, all the functions involved must also be differentiable.

In your example, if f(g(x)) is differentiable, then f(x) and g(x) must both be differentiable in order for the derivative to exist. This is a fundamental property of differentiability and is essential in understanding how composite functions behave.

I hope this helps clarify your question. If you have any further questions, please don't hesitate to ask.

Best,
 

1. What is a composite function?

A composite function is a function that is made up of two or more functions. It is represented by f(g(x)), where g(x) is the inner function and f(x) is the outer function. The output of the inner function is used as the input for the outer function.

2. How do you determine if a composite function is differentiable?

A composite function is differentiable if both the inner and outer functions are differentiable at the point of interest. This means that the limit of the composite function as x approaches the point of interest must exist and be equal to the value of the composite function at that point.

3. What is the Chain Rule in regard to differentiability of composite functions?

The Chain Rule is a calculus rule that allows us to find the derivative of a composite function. It states that the derivative of a composite function f(g(x)) is equal to the derivative of the outer function f'(x) evaluated at the inner function g(x), multiplied by the derivative of the inner function g'(x).

4. Can a composite function be differentiable at a point but not continuous?

Yes, it is possible for a composite function to be differentiable at a point but not continuous. This can happen when the inner function is not continuous at that point, but the outer function "fixes" the discontinuity caused by the inner function.

5. Why is the differentiability of composite functions important in mathematics and science?

The concept of differentiability of composite functions is important in mathematics and science because it allows us to analyze and model complex systems by breaking them down into simpler functions. It also allows us to find the rate of change of a system at a specific point, which has numerous applications in fields such as physics, engineering, and economics.

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