For $f:\mathbb{R}^{n} \to \mathbb{R}$, for which $\alpha > 0$ does the condition $|f(x)| \leq |x|^{\alpha}$ imply $f$ is differentiable at $0$ and why?
I have seen a solution to this question and it is similar to mine, but I want to understand what was faulty with my reasoning compared to the correct solution.
My Attempt:
We are trying to find under what conditions $\alpha > 0$ would imply my function is differentiable at $0$. TO be differentiable at $0$ means there exists a value $Df(0)$ such that:
$$\lim_{h \to 0} \frac{f(0 + h) - f(0) - D(f(0) \cdot h}{|h|}\ \text{where}\ h = |x-a|$$
Observe: $|f(0)| \leq |0|^{\alpha}$
As such we break things up into cases:
Case 1: if $f(0) = 0$. This means
$$\lim_{h \to 0} \frac{|f(h)|}{|h|} \leq \frac{|h|^{\alpha}}{|h|} \\ \Rightarrow\ \lim_{h \to 0} |h|^{\alpha - 1} = 0$$
So we can conclude that when $f(0) = 0$ that as long as $\alpha > 1$, then our function will be differentiable.
Case 2: $f(0) \neq 0$.[This is where my approach falls apart]. Again we have the condition:
$$\lim_{h \to 0} \frac{f(0 + h) - f(0) - D(f(0) \cdot h}{|h|} $$
But my concern is that since we have the observation that $|f(0)| \leq |0|^{\alpha}$ then
$$\lim_{h \to 0} \frac{|f(0 + h) - f(0) - D(f(0) \cdot h|}{|h|} \geq \frac{|f(h)|}{|h|} $$
My version of Case 1, is pretty much in line with what the correct solution said, except for considering $f(0) = 0$. Should I not have looked at it in terms of $f(0) = 0$?
How could I reconcile this?
Edit: I think I just realized that the issue is that I overlooked the idea that I am considering the Norm of the function, which means that whatever concerns I had were actually captured in my Case 1. Let me know if this is the right reasoning.
