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I am starting with Mathematica and I am willing to use it to perform some cumbersome computations for me.

Specifically, I want to obtain the duals of some convex cones and that requires extending some well-known functions. For instance, the functions

$$y\exp \left(\frac{x}{y} \right) \text{ and } x\log \left(\frac{x}{y} \right)$$

are of my interest and I would like to define $0\exp(\frac{x}{0})=0$ for each $x$ and $0\log(0)=0$, $0\log(\frac{0}{0})=0$, and $x\log(\frac{x}{0})=\infty$.

Can anyone help me to work this out?

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There are several ways to accomplish your goal, if I understand it correctly. For instance,

Clear[f];
f[x_, 0] := 0
f[x_, y_] := y Exp[x/y]

f[1, 1]
(* E *)

f[1, 0]
(* 0 *)

or

Clear[g]
g[x_, y_] := If[y == 0, 0, y Exp[x/y]]

g[1, 1]
(* E *)

g[1, 0]
(* 0 *)

The second function in the question can be handled similarly.

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It seems that what you are interested in is how to attach 'special' definitions for a given function. By this I mean the need to allow eg $y e^\frac{x}{y}$ evaluate normally for all values of $x$ and values of $y$ other than $0$ while making it evaluate to $0$ whenever $y$ is $0$ (the other function can be treated in principle with the same approach so it won't be discussed further).

There are more than a handful of ways to achieve such an effect in MA/WL, two of which-the most straightforward IMHO-I'll display in what follows.

The first way to achieve the desired effect is to construct a function with different 'branches' for the different cases you are interested in.

To that effect you can use Piecewise:

f[x_,y_]:=Piecewise[{
  {y e^(x/y), y!=0},
  {0, y==0}},
  $Failed
 ]

A couple of notes on f:

Inputs are assumed to be numerical quantities. This is implicit but could be defined more rigorously if that is desired (potential ways to do that with minor adjustments in f would include PatternTest's on the inputs or incorporating numeric checks (eg using NumericQ) within Piecewise branches-there are still more ways to achieve this eg using Condition).

Related to this is that f will return $Failed with symbolic inputs for y. More rigorous (and probably more demanding) definitions of f could involve Attribute NumericFunction but this is probably excessive for the current context.

Finally, similar results can be obtained with Which.

The second way to achieve similar effects with f above, is to make use of how MA/WL uses/matches patterns.

When evaluating an expression, all known definions are applied on the input until it no longer changes (this is a simplified way to put it but it suffices for the task at hand).

MA/WL uses specific definitions before more general ones during evaluation. To that effect g[x_,0]=0 is more specific than g[x_,y_]:=y Exp[x/y].

Evaluating these definitions will let MA/WL 'know' that expressions like g[_,0] are more specific than expressions like g[a,b] so when it encounters patterns like the former, it will supply the 'correct' replacement ie $0$ instead of the full definition of $y e^\frac{x}{y}$.

Obviously, numeric constraints and similar considerations to those presented in the first approach are relevant to this approach, too.

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