Can the Income Consumption Curve have a non smooth shape? Can the Income Consumption Curve have a shape that is kinked?
An easy example:
When the utility function is
$$
U(x,y) = \ln x + y
$$
and the prices are
$p_x = p_y = 1$, the IOC is kinked at income level $I=1$:
$$
(x,y) = \left\{
\begin{array}{ll}
\left( I, 0 \right) &amp; \text{ if } I \leq 1 \\
\left( 1, I-1 \right) &amp; \text{ if } 1 &lt; I
\end{array}
\right.
$$

A more complicated example:
When the utility function is
$$
U(x,y) = \min\left(2x+2y,x+10\right)
$$
and the prices are
$p_x = 8$, $p_y = 6$,
the good $y$ exhibits inferior (and Giffen) behavior. The IOC is kinked at income levels 30 and 80:
$$
(x,y) = \left\{
\begin{array}{ll}
\left( 0, \frac{I}{6} \right) &amp; \text{ if } I \leq 30 \\
\left( \frac{2I}{10} - 6, 8 - \frac{I}{10} \right) &amp; \text{ if } 30 &lt; I \leq 80 \\
\left( \frac{I}{8}, 0 \right) &amp; \text{ if } 80 &lt; I
\end{array}
\right.
$$