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These are not h/w problems but something from the class notes which I am not able to fully understand. I have two questions stated below

__Question1__We are doing chain rule and function of several variables. To explain the prof has first explained about single variables and then gone to composite functions and vector functions. But since I cannot understand single variables, no point in going to other topics

Anyway the class note is as follows:

**The differentiability of f at xo is equivalent to**

( f(xo+h) - (f[tex]^{'}[/tex](x0)h) ) / h -> 0 i.e. the function f is approximated to first order by function x-> f(xo) + f[tex]^{'}[/tex](x0)(x-xo) (where x = xo + h)

The function h-> f(xo) + f[tex]^{'}[/tex](x0)h is a function of the form: a constant function (f(xo) + a linear map (f[tex]^{'}[/tex](x0)h). Such a function is called an affine function. Thus differentiability can be formulated as the statement: there is a linear map [tex]\phi[/tex]: R->R such that the affine function f(xo) + [tex]\phi[/tex](x-xo) is a first-order approximation to f at xo.

( f(xo+h) - (f[tex]^{'}[/tex](x0)h) ) / h -> 0 i.e. the function f is approximated to first order by function x-> f(xo) + f[tex]^{'}[/tex](x0)(x-xo) (where x = xo + h)

The function h-> f(xo) + f[tex]^{'}[/tex](x0)h is a function of the form: a constant function (f(xo) + a linear map (f[tex]^{'}[/tex](x0)h). Such a function is called an affine function. Thus differentiability can be formulated as the statement: there is a linear map [tex]\phi[/tex]: R->R such that the affine function f(xo) + [tex]\phi[/tex](x-xo) is a first-order approximation to f at xo.

My problem is as follows

- it is the last line which states that differentiability can be formulated by the linear map. Can someone explain what the notes are trying to say. I have been on this for about 3 days and not able to grasp it

__Question 2:__Not entirely related to this, but assuming you have a function f(x1,x2,x3) = x1x2x3 + x1*x3. How do I prove that it is continuously differentiable. If I prove that partial derivatives exist, will it prove that they are continuously differentiable.

Thanks

Asif