Graph of the Rounding Function $y = \operatorname{round}(x)$

$y = \operatorname{round}(x)$ rounds $x$ to the nearest integer. If the fractional part is below $0.5$ it rounds down, and above $0.5$ it rounds up.

• $\operatorname{round}(2.3) = 2$
• $\operatorname{round}(2.6) = 3$
• $\operatorname{round}(-1.2) = -1$

Domain and range. The domain is all real numbers and the range is all integers. The graph is a staircase of horizontal segments of width $1$, each centered on an integer. For example, the value is $0$ on $-0.5 < x < 0.5$ and $1$ on $0.5 < x < 1.5$.

Discontinuities and jumps. The function is discontinuous at the half-integers $x = n + 0.5$, where the value jumps up by $1$. How these exact midpoints are rounded is fixed by a convention. The most common 'round half up' rule gives:

• $\operatorname{round}(0.5) = 1$
• $\operatorname{round}(2.5) = 3$

In statistics and accounting, round half to even (banker's rounding) is often used to reduce bias, and the treatment of negatives and midpoints varies between implementations.

Symmetry. Away from the half-integers, $\operatorname{round}(-x) = -\operatorname{round}(x)$, so the function is essentially odd (point-symmetric about the origin). Only at the midpoints can the tie-breaking rule spoil this symmetry.

Relation to other functions. Under the round-half-up rule it can be written with the floor function:

So rounding is a sibling of the floor and ceiling functions, choosing whichever of $\lfloor x \rfloor$ and $\lceil x \rceil$ is closer to $x$.

Applications. Rounding is used wherever continuous values must be reduced to convenient integers: displaying amounts of money, matching the number of digits of a measurement, and quantizing digital signals.