Differentiability implies continuous derivative?

In summary, we discussed the implications of differentiability and continuity in multi-variable functions, noting that in the case of one independent variable, it is possible for a function to be differentiable throughout an interval but with a non-continuous derivative. We gave examples of such functions and discussed the special conditions needed for a function to be differentiable at all points. We also mentioned the existence of functions that are continuous but nowhere differentiable or differentiable only at finitely many points. Finally, we discussed Darboux's theorem, which states that the derivative of a function satisfies the intermediate value property.
  • #1
kelvin490
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We know differentiability implies continuity, and in 2 independent variables cases both partial derivatives fx and fy must be continuous functions in order for the primary function f(x,y) to be defined as differentiable.

However in the case of 1 independent variable, is it possible for a function f(x) to be differentiable throughout an interval R but it's derivative f ' (x) is not continuous?
 
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  • #2
Yes, it is. I gave an example, [itex]f(x)= x^2 sin(x)[/itex] if x is not 0, f(0)= 0, when you asked this question on another board.
 
  • #3
HallsofIvy said:
Yes, it is. I gave an example, [itex]f(x)= x^2 sin(x)[/itex] if x is not 0, f(0)= 0, when you asked this question on another board.

I think you meant ## x^{2}sin(1/x)## when x is not zero.
 
  • #4
kelvin490 said:
However in the case of 1 independent variable, is it possible for a function f(x) to be differentiable throughout an interval R but it's derivative f ' (x) is not continuous?
The simplest such function is f(x) = |x| on [-1, 1] (I know, there is a problem with f'(0), but let f'(0)=0 and see what happens).

A lot of functions with discontinuous derivatives can be found by starting with a discontinuous function g(x) and let [itex]f(x) = \int_{0}^{x}g(t) dt [/itex].
 
  • #5
Svein said:
The simplest such function is f(x) = |x| on [-1, 1] (I know, there is a problem with f'(0), but let f'(0)=0 and see what happens).

A lot of functions with discontinuous derivatives can be found by starting with a discontinuous function g(x) and let [itex]f(x) = \int_{0}^{x}g(t) dt [/itex].

The function ##f(x)=|x|## is not differentiable at ##x=0##. For the second example, the function ##f## is usually not differentiable everywhere, one can only prove that it is differentiable almost everywhere. Fro the function ##f## in this example to be differentiable at all points, the function ##g## has to be very special.

The simplest example of the function differentiable everywhere with non-continuous derivative is probably the example by HallsofIvy (with the correction by lavina)
 
  • #6
Hawkeye18 said:
For the second example, the function ff is usually not differentiable everywhere, one can only prove that it is differentiable almost everywhere. Fro the function ff in this example to be differentiable at all points, the function gg has to be very special.
What happened to the old rule: "To derive with respect to the upper limit of an integral, insert the upper limit into the integrand".
 
  • #7
Svein said:
What happened to the old rule: "To derive with respect to the upper limit of an integral, insert the upper limit into the integrand".
As quoted this rule work only for continuous integrands. In the general case it only guaranties equality almost everywhere.
 
  • #8
Svein said:
The simplest such function is f(x) = |x| on [-1, 1] (I know, there is a problem with f'(0), but let f'(0)=0 and see what happens).

A lot of functions with discontinuous derivatives can be found by starting with a discontinuous function g(x) and let [itex]f(x) = \int_{0}^{x}g(t) dt [/itex].

There are also functions that are continuous but nowhere differentiable or differentiable only at finitely many points.
There are functions that are differentiable except on a Cantor set.
 
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  • #9
As others have pointed out, a differentiable function need not have a continuous derivative. However, the derivative ##f'## does satisfy the "intermediate value property," which means that if there are points ##x## and ##y## with ##f'(x) = a## and ##f'(y) = b##, then for any given ##c## between ##a## and ##b##, there is some point ##z## such that ##f'(z) = c##. This fact is known as Darboux's theorem and is interesting because it implies that ##f'## cannot have a step discontinuity. This does not contradict the example given by HallsOfIvy, in which ##f'## has an "oscillate to death" discontinuity, not a step discontinuity.
 
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  • #10
It took me absurdly long to realize how trivial Darboux's theorem is: if f' > 0 at some point and < 0 at another point then f cannot be monotone in between (by the intermediate value theorem), so f has a local extremum where f' = 0.by the way @Kelvin, the continuity of the partials is sufficient but not necessary for differentiability in several variables. the definition of differentiability is in terms of an approximating linear function.
 
  • #11
To maximize the confusion: Let I = [0, 1] and J be an open interval such that I ⊂ J. Then there exists a function that is C, equal to 1 on I and equal to 0 outside J.
 

FAQ: Differentiability implies continuous derivative?

1. What is the definition of differentiability?

Differentiability is a mathematical concept that describes the smoothness of a function at a particular point. A function is considered differentiable at a point if it has a well-defined derivative at that point.

2. How is differentiability related to continuity?

Differentiability implies that a function is continuous at a particular point. This means that the function is defined and has a limit at that point, and the value of the function at that point is equal to the limit.

3. What is the significance of having a continuous derivative?

A continuous derivative means that the function is smooth and has no sudden changes or breaks in its slope. This is important in many fields, such as physics and engineering, where the behavior of a system is described by a function and its derivatives.

4. Can a function be continuous without being differentiable?

Yes, it is possible for a function to be continuous but not differentiable at certain points. This can occur when there is a sharp corner or discontinuity in the function, which prevents it from having a well-defined derivative at that point.

5. How can we determine if a function is differentiable?

A function is differentiable if it satisfies the definition of differentiability, which states that the limit of the difference quotient (hence the slope of the secant line) approaches a finite value as the interval between two points approaches zero. This can be determined by evaluating the limit or by using the rules of differentiation.

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