# Continuity problem 2

• Oct 2nd 2010, 04:28 PM
shannu82
Continuity problem 2
Suppose that the function f and g are discontinuous at a. Which of the following are true...
i) f+g can never be continuous at a.
ii) 2f can never be continuous at a.
iii) f/g can never be continuous at a.
If it is true give a valid proof, if its false then give a counter example.

Any help would be appreciated. I believe i and ii is true but iii is false but don't quite understand how to show the proof...
• Oct 2nd 2010, 05:21 PM
zzzoak
If
f=0 x<0
f=1 x>=0

g=1 x<0
g=0 x>=0

f+g=1 everywhere.
• Oct 2nd 2010, 07:04 PM
Jhevon
Quote:

Originally Posted by shannu82
Suppose that the function f and g are discontinuous at a. Which of the following are true...
i) f+g can never be continuous at a.
ii) 2f can never be continuous at a.
iii) f/g can never be continuous at a.
If it is true give a valid proof, if its false then give a counter example.

Any help would be appreciated. I believe i and ii is true but iii is false but don't quite understand how to show the proof...

(i) and (iii) are false. zzzoak gave you a counter example for (i). try to find one for (iii) -- Hint: i'd let f = g = ..., that is, make f and g the same function. Play around with that. To prove (ii) is true, go back to the definition of what it means to be continuous: $\displaystyle \displaystyle \lim_{x \to a} f(x) = f(a)$, provided $\displaystyle \displaystyle f(a)$ is defined and all that carry on... personally, i'd use a proof by contradiction, or contrapositive, if that tickles your fancy.