Continuity and derivability of a multivariable function at a specific point

A good question for learning purpose. Let us somewhat refine the definition of the function under consideration:

f[x_, y_] := Sqrt[RealAbs[x^2 - x*y]]

We start from the verification of its continuity at the origin through

Limit[f[x, y], {x, y} -> {0, 0}] == f[0, 0]
(*True*)

Next, in order to test its differentiability at the origin, we find its partial derivatives at the origin by

Limit[(f[x, y] /. y -> 0 - f[0, 0])/x, x -> 0]
(*Indeterminate*)

and the analogous limit for the partial derivative by $y$. Since the partial derivative by $x$ does not exist at the origin, the function under consideration is not differentiable at the origin.

Try another point on your own.

Addition. On the OP's request, let

f[x_, y_] := Sqrt[x^2 + y^2]*Cos[x*y]

Then

D[f[x, y], x] /. {x -> 0, y -> 1}
(*0*)

and

 D[f[x, y], y] /. {x -> 0, y -> 1}
(*1*)

Up to the definition of the differentiability , the condition

Limit[(f[x, y] - f[0, 1] + 0*(x - 0) - 1*(y - 1))/Sqrt[x^2 + y^2],{x, y}-> {0, 1}] == 0
(*True*)

should be tested in this case.

Edit. It should be

Limit[(f[x, y] - f[0, 1] + 0*(x - 0) - 1*(y - 1))/Sqrt[x^2 + (y - 1)^2], {x, y} -> {0, 1}] == 0
(*0*)

in the above. Sorry for the typo.