Final answer:
To establish the continuity of the function f(x, y) = x^2 / (x+y), the partial derivatives with respect to x and y are calculated, showing that the function is continuous wherever it is defined, i.e., for all (x, y) except where x+y=0.
Step-by-step explanation:
The question asks to show that the function f(x+y) = x^2 / (x+y) is continuous by using partial derivatives. This function, as stated, is not properly defined because f(x+y) suggests a function of a single variable, while the form x^2 / (x+y) indicates it's a function of two variables, particularly f(x, y) = x^2 / (x+y). Assuming that is the case, for f(x, y) to be continuous, its partial derivatives with respect to x and y must exist and be continuous on their domain, excluding points where x+y=0 since the function is not defined there.
Let's find the partial derivative with respect to x:
fx(x, y) = d/dx (x^2 / (x+y))
Applying the quotient rule for differentiation, fx(x, y) would be:
fx(x, y) = [(x+y)(2x) - x^2(1)] / (x+y)^2 = (2x^2 + 2xy - x^2) / (x+y)^2 = (x^2 + 2xy) / (x+y)^2
This partial derivative exists and is continuous except where x+y=0. Similarly, we would determine fy(x, y) and find it to also be continuous except where x+y=0.
Therefore, excluding the line x+y=0, the partial derivatives exist and are continuous, which implies that the function is differentiable and hence continuous on its domain.