Existence of limits and continuity

[sergey90]

Homework Statement

#1. If limit[x->a]f(x) exists, but limit[x->a]g(x) doesnt, limit[x->a](f(x)+g(x)) doesn't exist. T/F? (Proof or example please)

#2. prove that if f is continuous, then so is |f|

#3. f(x) = [[x]]+[[-x]] for what a does limit[x->a]f(x) exist? Where is f discontinuous?

Homework Equations

The Attempt at a Solution

#3 confuses me the most; my first thought is the function = 0 for all x so the limit should exist for all a and it should be 0, which means it should be continuous everywhere, but there is a theorem that states if f(x) is disc. on (a,b) and g(x) is disc. on (a, b) then f(x) + g(x) is disc. on (a,b) and since they are both disc. on integer values so should be their sum...so which thought is correct and why?

 

[LCKurtz]

but there is a theorem that states if f(x) is disc. on (a,b) and g(x) is disc. on (a, b) then f(x) + g(x) is disc. on (a,b)

What if f(x) = 0 if x rational and 1 if x irrational
and g(x) = 1 if x rational and 0 if x irrational?