N: The $\limsup_{ n \to \infty }{n}$ and $\liminf_{ n \to \infty }n$ define the two extremes of any space.

In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as discontinuities.

The epsilon–delta definition of a limit was introduced to formalize the definition of continuity.

In mathematics, the oscillation of a function or a sequence is a number that quantifies how much that sequence or function varies between its extreme values as it approaches infinity or a point.

Let $a_n$ be a sequence of real numbers. The oscillation $\omega (a_{n})$ of that sequence is defined as the difference (possibly infinite) between the limit superior and limit inferior of $a_{n}$:

$\omega(a_{n}) = \limsup_{ n \to \infty }a_{n} - \liminf_{ n \to \infty }a_{n}.$

The oscillation is zero if and only if the sequence converges. It is undefined if $\limsup_{ n \to \infty }$ and $\liminf_{ n \to \infty }$ are both equal to $+\infty$ or both equal to $-\infty$, that is, if the sequence tends to $+\infty$ or $-\infty$.

Examples:

• $1/x$ has oscillation $\infty$ at $x=0$, and oscillation $0$ at other finite $x$ and $+\infty$ and $-\infty$.
• $(-1)^x$ or $1, −1, 1, −1, 1, −1...$ has oscillation $2$.
• $\sin (1/x)$ (the topologist’s sine curve) has oscillation 2 at x = 0, and 0 elsewhere.