Question

Show that f=O(1) [this is the letter O not the number 0] as x->a if and only if f(x) is bounded on some neighborhood of a.

Answer 1

Question 1. Show that $f = O(1)$ as $x \to a$ if and only if $f(x)$ is bounded on some neighborhood of $a$.

Solution. Recall that $f(x) = O(g(x))$ as $x \to a$ iff there are $M > 0$ and $\delta > 0$, such that $|f(x)| \leq M |g(x)|$ for all $x$ with $|x - a| < \delta$. Use this definition in the case when $g(x) \equiv 1$ and obtain that there are $M > 0$ and $\delta > 0$ such that $|f(x)| \leq M$ for all $x$, such that $|x - a| < \delta$. This is precisely the statement that $f$ is bounded by $M$ in the $\delta$-neighborhood of $a$.