Thread: basic multivariable calculus question

I have been given the following question:

Find the domain of each function. Show that the function is $\displaystyle C^1$ (and is therefore differentiable) in this domain

The first function is $\displaystyle f(x,y) = \frac{x}{y}+\frac{y}{x}$

Obviously the domain is $\displaystyle \{(x,y):x \not = 0, y \not = 0\}$ and I can quite easily show the function is C1 by differentiating it and then using the theorems on continuity when adding and dividing continuous functions. But the phrase "and is therefore differentiable" is bothering me: they seem to be implying that we can use this to prove differentiability instead of using the derivative and I'm worried that I have forgotten something very basic that should be used.

So if one of the clever people on this site could tell me what I've forgotten (or even just a name so I can look it up) that would be wonderful. And if they can tell me I have it right already, even better.

I remember reading something like "a function is differentiable in the particular interval if it is continuous in that interval".

Thanks for replying, Altair, but I think what you read should have been "a function is continuous in the particular interval if it is differentiable in that interval" ie. differentiable implies continuous and not the other way round. For example, the function y = |x| is continuous but not differentiable.