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    Can anyone give me starting to point to prove that a linear map f is differentiable everywhere when it goes from Real N to Real M and f(x) is an mxn matrix acting on a vector x?

    I can't see how to show something is differentiable?

    What I've looked up online talks about smooth functions - what's a smooth function?

    Also does anyone have a simple guide/definition and/or example on how to determine the derivative of a vector function from one dimension to another and also what a jacobian matix is?


    Sorry for all the questions... I'm trying to catch up with everything so I don't get dragged down for the exams.
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    I've found a similar question in a past exam where they state the function is differentiable as all the partial derivatives are continuous. Is there a theorem that backs up this reasoning?
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    The gradient of a function f: \mathbb{R}^m \to \mathbb{R} ^n, denoted by \nabla f(x), is the mxn matrix (\alpha _{ij} ) = \partial _j f_i , where \partial _j f_i = \frac{\partial f _i}{\partial x_j}. This matrix is called the Jacobian matrix.

    Smooth function: http://mathworld.wolfram.com/SmoothFunction.html
 
 
 
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