Can budget lines be discontinuous? It's been awhile since I've taken introductory microeconomics. I read budget lines can be discontinuous awhile ago but am suspicious as to such a concept being discussed in intro micro classes particularly calling them budget 'lines'.



The problem:


  Say Jill likes food and clothing. Jill has 200 to spend on food and clothing. One unit of food costs \$1 while one unit of clothing costs $2.


Budget line is 200=F+2C


  Say Jill receives an 80% discount on food but it applies up to 100 units.


New budget line is 200=0.2F+2C for 0-100 then same as old for 100 onward?

So it's discontinuous? So it's not really a budget 'line' right (though piecewise, it's a line, I guess).
I guess the discount holds for the 100 first units of food even if she buys more than 100. Therefore the new budget line is
\begin{equation*}
\left\{
   \begin{array}{ll}
   &amp; 200 = 0.2 F + 2C \text{ if } F &lt; 100 \\
   &amp; 200 = 0.2 \times 100 + F-100+2C \text{ otherwise }
   \end{array}
\right.
\end{equation*}
which is continuous.