For all $x \neq y$ we have:
$$(f(x) - f(y))^2 + (g(x) - g(y))^2 \geq \frac{1}{2} (|f(x) - f(y)| + |g(x) - g(y)|)^2 > \frac{1}{2}$$
For every $x$, consider the point $(f(x), g(x))$ in the $\mathbb{R}^2$ plane. The above inequality shows that the distance between every two of these points is more than $\frac{1}{\sqrt{2}}$. For every point $(f(x), g(x))$, consider a circle with center at this point and a radius of $\frac{1}{2\sqrt{2}}$, so not two of these circles intersect. On the other hand the number of these points are uncountable, so we have an uncountable number of circles in the plane that no two of them intersect and this is impossible; because there is a point with rational coordinates in every one of the circles, hence we would have an uncountable number of points with rational coordinates and since we know that the total number of the points with rational coordinates is countable, this is a contradiction; so no two functions with this property exist.