What is the term used to describe a discrete function which is non-zero at only 1 point, and zero everywhere else? Intended for a spatial domain.

Question

Please see the image below.

What is the formal term for the type of function shown below.

• One could describe it as a uniform distribution and the domain being x=5. But this is not elegant
• One could describe it as a logic signal at x=5, but this is not very accurate.

It is intended for a spatial domain. So if there is a grid, I would like to describe a function which is "a" at only one point in the grid, and "b" everywhere else.

Is a succinct term for this function?

Answer 1

Perhaps you want the Kronecker delta: $$ \delta_{5x} = \begin{cases} 1 \quad x = 5 \\ 0 \quad \text{otherwise} \end{cases} $$

Edit in response to an edit in the question.

This will work for a grid: $\delta_{xy}$ makes sense when $x$ and $y$ are elements of any universe $S$ so you can write $$ f_x(y) = (a-b)\delta_{xy} + b . $$