(a) Let be a function, and let . Define what it means for to be continuous at . Show that is continuous at if and only if for every sequence with .
(b) Let be a non-constant polynomial. Show that its image is either the real line , the interval for some , or the interval for some .
(c) Let , let be continuous, and assume that holds for all . Show that must be constant.
Is this also true when the condition that be continuous is dropped?