This question is straight from Spivak Chapter 9, Question 15. I have attempted the previous problem, which is another differentiable function I had to prove.
I want to attempt this problem using the definition of a derivative as a limit. The issue in this problem for me is the $\leq$ sign. How do you work with this restriction? Any hints are appreciated.
Surely I can take $f$ to be $x^2$ since this satisfies the condition that $|f(x)| \leq x^2$? Is this sufficient to prove the condition?
Proof. Let a function $f$ be defined as $|f(x)| < x^2$. Since $x^2$ is such a function, we know that the derivative of $x^2 = 2a$, and since $x^2$ is differentiable at 0, the (half-assed) proof is complete.
I have skipped many steps in the proof. I merely asserted that $x^2$ is differentiable at 0 and meets the restriction specified.