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$$u(x, y) = 2$$
is a simple example of a continuous utility function with thick indifference curve.
\begin{eqnarray*} u(x, y)= \begin{cases} x & x < 1 \\ 1 & 1 \leq x < 2 \\ x - 1 & x \geq 2 \end{cases} \end{eqnarray*}
is another continuous utility function with thick indifference curve for $u=1$.
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