Making $g(x)=x^2\operatorname{sgn}(x)$ continuous at $0$. How would the function $g(x) = x^2 \operatorname{sgn} (x) $ be defined at $x= 0 $ so that it is continuous there?
This makes no sense to me. It isn't, period. Why would it be? I can't re-define the function and $\operatorname{sgn}(x) = \frac{0}{|0|} $ at $x=0$.
What is being asked here?

 A: The sign function is actually not defined as you say for $x=0$. It may be equal to $0$, for symmetry reasons, among other reasons. But let's come back to you question. The limits $$\lim_{x\to0, x<0}x^2\mathrm{sgn}(x)=0$$
and $$\lim_{x\to0,x>0}x^2\mathrm{sgn}(x)=0$$
are equal, so taking the value $g(0)=0$ is fine for continuity. No other value would work of course.
As for differentiability, check if the limits
$$\lim_{x\to0,x<0}\frac{g(x)-g(0)}{x-0}\;\text{and}\;\lim_{x\to0,x>0}\frac{g(x)-g(0)}{x-0}$$
exist and are equal, in which case, $g$ would be differentiable at $x=0$.
A: Sometimes in analysis, we look at functions which we would really like to posses some nice properties, but don't. Sometimes, we can 'fix' these functions by looking at ones that are identical to it almost everywhere.
In your case, $\text{sgn}(x)$ (the sign function) is defined for all nonzero real numbers, since those are the only ones with a sign. i.e., $\text{sgn}(x)$ maps: $\mathbb{R}\backslash\{0\}\rightarrow \mathbb{R}$. This is not good news at all because we like dealing with functions that are defined over the entire space, and this one isn't.
The question is asking:

How should we extend $x^2\cdot \text{sgn}(x)$, a function from $\mathbb{R}\backslash\{0\} \rightarrow \mathbb{R}$, to another function from $\mathbb{R}\rightarrow\mathbb{R}$ so that the resulting function is continuous along $\mathbb{R}$?

Basically, you are trying to find the nicest way to extend your original function to the whole space. Extending it so that $g(0)=-5093847e^{i\frac{\pi}{4}}$ would not make the function very nice to work with. Which extension makes the function continuous?
As you are seeing in your study of analysis, continuous functions posses properties that make them easy to work with and gain insight from, so there is motivation to try and make things continuous (or better, differentiable) when you can.
