Can budget lines be discontinuous?

Question

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'.

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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).

Answer 1

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} & 200 = 0.2 F + 2C \text{ if } F < 100 \\ & 200 = 0.2 \times 100 + F-100+2C \text{ otherwise } \end{array} \right. \end{equation*} which is continuous.