The first form can be instead expressed as: $$f(x) - c\lfloor \frac {f(x)}c \rfloor$$ From the form of the rewritten expression, it is clear that subtracting $-c\lfloor \frac {f(x)}c \rfloor$ will render it continuous. The second form is a bit more interesting. f(x mod c) is a function where t...