Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Given the inequalities:

$$|f(x) - g(x)| < \epsilon\quad \forall \quad x \in [a,b]$$

and

$$|g(x)| < M \quad \forall \quad x \in [a,b]$$

where $\epsilon > 0$ and $M > 0$.

What is the tightest bound that I can get on $|f(x)|$

share|improve this question

1 Answer 1

up vote 2 down vote accepted

Triangle inequality gives us

$$|f(x)| \leq |f(x) - g(x)| + |g(x)| < \epsilon + M$$

share|improve this answer
    
Thanks! (also: facepalm) –  Elements Nov 15 '12 at 4:42
    
Take for instance $f(x) = x+ \epsilon$, $g(x) = x$. Then we would get above. If additional properties are imposed on f,g we could get tighter bounds. –  Gautam Shenoy Nov 15 '12 at 4:44
    
Was the "facepalm" because it was obvious or because you expected something better? –  Gautam Shenoy Nov 15 '12 at 4:46
    
No not at all! This was exactly what I was looking for actually. I just spent a lot of time going in circles trying to show it. –  Elements Nov 15 '12 at 4:49
1  
Your welcome then. –  Gautam Shenoy Nov 15 '12 at 4:50

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.