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How can a function be differentiable if its partial derivatives are not continuous?

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- Thread starter Jazzyrohan
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- #1

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How can a function be differentiable if its partial derivatives are not continuous?

- #2

member 587159

Consider ##f(t) = t^2 \sin(1/t)## when ##t \neq 0## and ##f(0) =0##. This function is differentiable on ##\mathbb{R}## but its derivative isn't continuous.

Why would the result hold in the higher derivative case? (Similar counterexamples are possible)

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- #4

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For a real-valued function of two variables, the equivalent definition is that the difference ##f(\mathbf{x}+\mathbf{\Delta x}) - f(\mathbf{x})## can be similarly approximated with a dot product ##\mathbf{f'(x)}\cdot\mathbf{\Delta x}##. Here the derivative ##\mathbf{f'(x)}## is now a two-component vector and it is the same as the gradient of the function ##f(\mathbf{x})##.

If the function takes two arguments and returns a two-component vector, the derivative ##\mathbf{f'(x)}## is a ##2\times 2## matrix and the differential is ##\mathbf{f'(x)}\mathbf{\Delta x}##, where there's a matrix-vector multiplication instead of a dot product.

- #5

fresh_42

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Maybe the beginning of https://www.physicsforums.com/insights/the-pantheon-of-derivatives-i/ can help you here. As @hilbert2 has said: differentiability is only the possibilty to approximate a function locally by a linear function. The question, whether these many local events result in a continuous dependency of the location is a completely different one. The most common mistake comes from the notation: ##f\,'(x)##. It is wrong. It should better be ##f\,'(a)##, since this is what differentiation does: it gives a slope at a certain point. Differentiability now means, that this can be done at all points ##x=a##, such that we get a new function ##a \longmapsto f\,'(a)## which people write ##x \longmapsto f\,'(x)## and since people are lazy, abbreviate it by ##f\,'(x)##. The result is, that the dependency of the location, at which ##x \longmapsto f(x)## has been approximated by a linear function via the calculation ##\left.\dfrac{d}{dx}\right|_{x=a}f(x)## is completely lost.

So what differentiablity means is that those linear approximations exist, at a point or everywhere.

Now whether this differentiabilty depends continuously or differentiable from said location is another step.

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- #7

Stephen Tashi

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,what is the basic condition for such a function to be differentiable?

Are you asking for

- #8

fresh_42

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I didn't want to criticize you. My post has been meant as a general reminder, because there are so many different ways to regard a differentiation, and that short notations often disguise the individual view. I once listed a few views of it and found ##10##: first page here https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/ and the word "solpe" didn't even occur in the list. It was fun to see how a simple tangent can have so many roles.

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Can you also tell me some books containing good theory or visualization and a few books for problem solving?I am currently in first year of college.

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