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Jazzyrohan

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

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In summary, the author has been studying multivariable calculus but can't quite think visually how a function will be differentiable at a point. Consider a function ##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? Got it.

- #1

Jazzyrohan

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

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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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Jazzyrohan

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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.

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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 possibility 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.Jazzyrohan said:

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

Are you asking for

- #8

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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.hilbert2 said:

- #9

Jazzyrohan

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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.

The differentiability of a function of two variables refers to the ability of a function to have a well-defined tangent plane at any given point in its domain. In other words, the function must have a unique slope and direction at that point.

A function must be continuous in order to be differentiable. This means that the function must have no abrupt changes or "jumps" in its graph. However, not all continuous functions are differentiable.

Partial derivatives are the derivatives of a function of multiple variables with respect to a single variable, while total derivatives take into account the effect of all variables on the function. Total derivatives can be found by taking a linear combination of the partial derivatives.

One way to determine differentiability is to check if the function satisfies the necessary and sufficient conditions for differentiability, such as having continuous partial derivatives at a specific point. Another method is to calculate the partial derivatives at a point and then use the definition of the total derivative to check for differentiability.

No, a function cannot be differentiable at a point if it is not continuous at that point. This is because differentiability implies continuity, and if a function has a "jump" or abrupt change at a point, it does not have a well-defined tangent plane and therefore is not differentiable at that point.

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