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it is known that by using delta-functions we can somehow assign a meaning to "functions" that take a specific value at one point, as a sort of generalization of the Kronecker delta-function in discrete domain.

Instead I would like to consider the domainR^{2}, and formally define a "function" that takes only specific values along aone-dimensional pathinsideR^{2}. This path on R^{2}is defined by the points [itex]\left( x,p(x) \right)[/itex], with p:R--->R.

Note: if I had had to define a function taking specific values at some specific points, I would have defined it as a finite sum of delta functions [itex]\delta (x,y)[/itex], but in this case I cannot use discrete sums of delta functions.

Is it possible to express formally such a (non-)function?

Is it allowed to do something like the following:

[tex]\int_{R^2}f(\mathbf{t})\delta(\mathbf{t}-\mathbf{p})|d\mathbf{t}| = f(\mathbf{p}) = f(x,p(x))[/tex]

where the pathpis given by [itex]\mathbf{p}(x)=\left( x,p(x) \right)[/itex] ,tis some position vector in [itex]R^2[/itex], and [itex]\delta[/itex] is the Dirac delta-function in 2 dimensions.

Thanks!

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# Two-variable functions defined on a one-dimensional subset

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