For the case of two goods, give an example of preferences that are represnted by a continuous utility function that allows for fat indifference curves The question in the title sounds like a trick question, due to the monotonicity property that indifference curves have, such that for two goods x and y, strong monotonicity implies y > x. 

Possible that you could answer the question by stipulating the area under a curve?
$$u(x, y) = 2$$
is a simple example of a continuous utility function with thick indifference curve.

\begin{eqnarray*} u(x, y)= \begin{cases} x &amp; x &lt; 1 \\ 1  &amp; 1 \leq x &lt; 2 \\ x - 1 &amp; x \geq 2 \end{cases} \end{eqnarray*}

is another continuous utility function with thick indifference curve for $u=1$.