I am having a hard time understanding the concept of differentiability with functions that take in a vector input. For example, we have a function f(x) that is defined as the norm of vector x.
This is a scalar valued function as it will give a real number as the result.
Firstly let us try to see if this function is continuous. For any vector x in 3-dimensional space, the norm will be a polynomial. Since polynomial functions are continuous, this function f(x) is also continuous.
Secondly , looking at partial derivatives. I looked up online that the partial derivative of the norm can be found out using the chain rule. This is where I got confused. I have seen many answers about this here on stackoverflow but I do not understand the reason behind getting the specific answer.
$$d\|{\bf x}\| = \frac{ {\bf x}^Td{\bf x}}{\|{\bf x}\|}$$
from link: Derivative of the $2$-norm of a multivariate function
It is not the chain rule that confuses me, it is more of the meaning behind this partial derivative. Our function takes in a vector say x and then spits out its norm. The partial derivative, as far as I understand, is the slope of the tangent along the direction we are taking the derivative in. Why are we getting a vector in the partial derivative?
Now coming onto proving whether the function is differentiable at x=0 or not. Again, I am aware that there are many answers available about this online. What I understand so far is that we could show a proof by contradiction by picking a point say for ex (t,0,0) and then plug it into the function. If we assume the function f(x) to be differentiable at 0, then plugging in the point (t,0,0) should also be differentiable at 0. But that would give us the mod function, which we know is not differentiable at 0. Thus we reach a contradiction.
However, I do not understand the intuition behind this. I tried using the differentiation formula using limits on this function, but I could not make sense of it.
I am sorry if this sounds very confusing, but I have spent a long time looking into this idea but have found various answers with various techniques without having much intuitive meaning. It would be great if someone could tie these concepts together! I would very much appreciate the help! Thank you!
