I am very aware that a partial derivative could be not continuous at a certain point. However, I don't understand when can a partial derivative doesn't exist. Because in the book, they always say "If the partial derivative exists …." or sometimes "If the partial derivative exists and is continuous at (x,y)…." So I assume, there must be a situation where it doesn't even exist
Also, unlike single variable calculus, where we know that some functions such as $|x|$ are not differentiable and thus we have no derivative. But I learned that (Hopefully I am correct) that partial derivative exists even when the function is not differentiable
However, for me, it is just about finding a derivative of a function, and I can just do that by following derivation rules so I can't think of a situation where I can't find a partial derivative of a function (I feel there should always be an expression regardless if that derivative is continuous at (x,y) or not.
I ask this question because when I was checking past calculus exams, there are always questions where they ask if the partial derivatives exist, while the function given is always in forms such as:
$$\displaystyle f(x,y) = \frac{x^2y^2}{x^4+y^4} \text{ that involves fractions}$$
Can someone give me an example and explain a bit for me?
NB: I am just learning multivariable calculus, so from what I have learned so far I can't think of any situation like this. Not sure if this happens in more advanced math.
Best Answer
Take, for instance,$$\begin{array}{rccc}f\colon&\Bbb R^2&\longrightarrow&\Bbb R\\&(x,y)&\mapsto&|x|.\end{array}$$Then $\frac{\partial f}{\partial x}(0,0)$ doesn't exist, since the limit$$\lim_{h\to0}\frac{f(h,0)}h\left(=\lim_{h\to0}\frac{|h|}h\right)$$doesn't exist.