# Quick and Simple Real Analysis Bound

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 $|f(x)|$

-

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