1. The problem statement, all variables and given/known data

2. The attempt at a solution
I'm not really sure where to start. We just want to show that ##\lim_{x \to c} \frac{f(x) - f(c)}{x - c} = 0##. I see that ##\lim_{x \to c} (x - c)^2 = 0##. I feel that this may be a simple trick of inequalities, but I am having a complete brain fart at the moment. Can anyone provide any direction? Thanks in advance for any response.

Aha, yes! I didn't even consider finding what ##f(c)## equals. Having ##|f(x)| \leq (x - c)^2## for all ##x \in I## forces ##f(c) = 0##. Now, to find ##f'(c)## we must now find $$\lim_{x \to c} \frac{f(x)}{x - c}$$, which we can show to be ##0##, if we can alternatively show that $$\lim_{x \to c}|\frac{f(x)}{x-c}| = \lim_{x \to c}\frac{|f(x)|}{|x-c|} = 0$$. Now, ##|f(x)| \leq (x - c)^2## for all ##x \in I##, also implies that for ##x \in I-\{c\}##, ##\frac{|f(x)|}{|x-c|} \leq |x - c|##. Since the latter's limit is 0, and both are point-wise positive, we have that $$\lim_{x \to c}|\frac{f(x)}{x-c}| = 0$$, and we are done.