\$...\$ gives \textstyle, but ... produces \displaystyle. You can see the difference if you set the formula \lim_{x\to0}\frac{f(x)}{g(x)} in the two...
Something has been bothering me too, ever since I saw this sketch a week ago. The configuration somehow looked impossible, but until now I could not...
The even extension is correct. The odd one is not, because the line segment from $(-2,-4)$ to $(0,-2)$ should actually go from $(-2,0)$ to $(0,-2)$....
For 2), divide the square into four smaller squares with side $1$. Any two points in the same smaller square will be within distance $\sqrt2$ of each...
Let $g(x) = f(x) - 0.5x$. Can you use the given information (together with the intermediate value theorem and Rolle's theorem) to show that $g'(c) =...