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Math Help - Continuity and Differentiability

  1. #1
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    Question Continuity and Differentiability

    I am confused about the definition of continuity and differentiability of n-variable functions. Also, what is the meaning of continuously differentiable?

    Here are some examples that I don't understand fully. Please explain.

    1. Let function f be defined on the whole xy plane. f(x,y)= 1 if x=y=/=0 and f(x,y)=0 otherwise. In this case, f is not continuous at (0,0) but both partial derivatives fx and fy exists at (0,0).

    2. Let function f be defined as f(x,y)=(x^1/3 + y^1/3)^3. f(x,y) is continuous and has partial derivatives at origin (0,0) but is not differentiable there.

    3. Let f be defined by f(x,y)=y^2 + x^3 sin (1/x) for x=/=0, and f(0,y)=y^2. f is differentiable at (0,0), but is not continuously differentiable there because fx(x,y) is not continuous at (0,0).

    Thanks. > <
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  2. #2
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    Let us start with the first one.
    Quote Originally Posted by catcat103 View Post
    1. Let function f be defined on the whole xy plane. f(x,y)= 1 if x=y=/=0 and f(x,y)=0 otherwise. In this case, f is not continuous at (0,0) but both partial derivatives fx and fy exists at (0,0).
    In single-variable calculus if f is differenciable at the point then it must be continous there. This example is showing that the existence of partial derivatives does not gaurentte the continuity of the function. Look at \partial_x f(0,0). By definition it is \lim_{h\to 0} \tfrac{f(h,0)-f(0,0)}{h} = \lim_{h\to 0}\tfrac{0}{h} = 0. Thus, \partial_x f(0,0) = 0. Similarly \partial_y f(0,0)=0. And so the partials exist.
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  3. #3
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    Also, what is the meaning of continuously differentiable?
    - differentiable
    - the derivative is continuous

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