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Given the inequalities:

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


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

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

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

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up vote 2 down vote accepted

Triangle inequality gives us

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

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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
Your welcome then. – Gautam Shenoy Nov 15 '12 at 4:50

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