# Big O notation

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More formally, if f and g are real valued functions of the real variable ${\displaystyle t}$ then the notation ${\displaystyle f(t)=O(g(t))}$ indicates that there exist a real number T and a constant C such that ${\displaystyle |f(t)|\leq C|g(t)|}$ for all ${\displaystyle t>T.}$
Similarly, if ${\displaystyle a_{n}}$ and ${\displaystyle b_{n}}$ are two numerical sequences then ${\displaystyle a_{n}=O(b_{n})}$ means that ${\displaystyle |a_{n}|\leq C|b_{n}|}$ for all n big enough.
The big O notation is also often used to indicate that the absolute value of a real valued function around some neighbourhood of a point is upper bounded by a constant multiple of the absolute value of another function, in that neigbourhood. For example, for a real number ${\displaystyle t_{0}}$ the notation ${\displaystyle f(t)=O(g(t-t_{0}))}$, where g(t) is a function which is continuous at t = 0 with g(0) = 0, denotes that there exists a real positive constant C such that ${\displaystyle |f(t)|\leq C|g(t-t_{0})|}$ on some neighbourhood N of ${\displaystyle t_{0}}$.