i got a big final exam tomorow, and im trying my best to understand all of the practice problems so i at least have a prayer. i have previous questions that im still trying to figure out, so any help on those would be wonderful.
3. Give examples of functions that satisfy the following properties, or explain why such functions do not exist.
Mention any theorems or results that you are using to justify your answers... and
justify your answers.
(a) A function f (x) that is differentiable in [-1; 1].
(b) A function f (x) that is continuous in [-1; 1] but not differentiable at some point in [-1; 1].
(c) A function f (x) that is integrable in [-1; 1] but is not continuous in [-1; 1].
(d) A function f (x) that is defined for every point in [-1; 1] but is not integrable in [-1; 1].
(e) A function f (x) that is defined for every point in [-1; 1] but is unbounded in that interval.
(f) A function f (x) that is continuous in [-1; 1] but is unbounded in that interval.
(g) A function f (x) that is differentiable in [-1; 1] but is unbounded in that interval.
a) any sort of polynomial is differentiable on any given interval
b) absolute value[x] is continuous on all real numbers, but is notdifferntiable at x=0 (i know it has a corner point there, but what would be sufficient reasoning for me to use on the test- i cant see corner point being enough for this teacher)
c) a step function, you can find the integral, however it is not continuous
d) i cant think of a specific function, but wouldnt just a random assortment of points that define every x work?
e) i dont think this exists, but i cant explain why
f) same as e
g) ?
im not too sure on some of these, especially the last 3, so if someone can help with those....also, i am not happy with my reasoning on some of these, so if anyone can tell me how i can improve...
i extremely appreciate anything that anybody can help me with