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athrun200
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Actually, even if the directional derivatives exist in all directions, that is still not a sufficient condition for differentiability.rjvsngh said:existence of partial derivatives does not imply "differentiability". in some sense, differentiability in higher dimensional spaces is a stronger condition than existence of partial derivatives. intuitively, partial derivatives only sample the function along "coordinate directions" but this is not enough to satisfy the condition of differentiability at a point because the function's behavior along coordinate directions may not represent it's behavior along other directions in higher dimensions.
From http://en.wikipedia.org/wiki/Differentiable_function#Differentiability_in_higher_dimensions", [itex]f(x,y) = \begin{cases}\frac{y^3}{x^2+y^2} \text{ if }(x,y) \ne (0,0) \\ 0 \text{ if }(x,y) = (0,0)\end{cases}[/itex].Bacle said:Any chance you (or anyone else) have an example, lugita15 ?
No, my example was to show that the existence of directional derivatives in all directions is still not sufficient to establish differentiability.LikeMath said:Recall that if the partial derivatives exist and continuous at (a,b) then the function is differentiable at (a,b),
your example does not contradict this fact.
lugita15 said:From this Wikipedia page, [itex]f(x,y) = \begin{cases}\frac{y^3}{x^2+y^2} \text{ if }(x,y) \ne (0,0) \\ 0 \text{ if }(x,y) = (0,0)\end{cases}[/itex].
Omega017 said:Can anyone explain why this function is not differentiable at (0,0)?
See here (pages 4-6).Omega017 said:Can anyone explain why this function is not differentiable at (0,0)?
If the partial derivatives exist and are continuous in some neighborhood of a point then the function is differentiable. Partial derivatives continuous at a point is not sufficient.LikeMath said:Recall that if the partial derivatives exist and continuous at (a,b) then the function is differentiable at (a,b),
your example does not contradict this fact.
Continuity and differentiability are two important concepts in calculus. Continuity refers to the smoothness of a function, meaning that there are no abrupt changes or gaps in the graph. Differentiability, on the other hand, refers to the existence of a derivative at a given point. In simpler terms, a function is continuous if it can be drawn without lifting your pencil from the paper, while a function is differentiable if it has a well-defined slope at a given point.
Yes, a function can be continuous but not differentiable. This means that the function has no abrupt changes or gaps, but it does not have a well-defined slope at a certain point. An example of such a function is the absolute value function, which is continuous but not differentiable at the point where the graph changes direction.
A function is differentiable if it has a well-defined derivative at every point in its domain. To determine if a function is differentiable, you can use the definition of a derivative or the differentiability rules, such as the power rule, product rule, and quotient rule. If a function satisfies these rules, it is considered differentiable.
Continuity and differentiability are essential concepts in calculus as they allow us to analyze the behavior of functions and solve problems related to rates of change. Continuity ensures that a function is well-behaved and can be easily studied, while differentiability allows us to find the slope of a function at a given point, which is crucial in optimization and curve sketching.
No, a function cannot be differentiable at a point but not continuous. This is because differentiability requires continuity at a given point. If a function is not continuous at a certain point, it cannot have a well-defined slope at that point, making it not differentiable. Therefore, a function must be continuous at a point to be differentiable at that point.