Word Count : 609

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We are asked whether the function $$ f(x,y) = \frac{x^2 - y^2}{x^2 + y^2}, \quad x \neq 0, y \neq 0 $$ satisfies Schwarz’s Theorem (equality of mixed partial derivatives) at the point $(1,1)$. --- ### Step 1: Recall Schwarz’s Theorem Schwarz’s (Clairaut’s) theorem states: > If $f_{xy}$ and $f_{yx}$ are continuous in a neighborhood of $(a,b)$, then > > $$ > f_{xy}(a,b) = f_{yx}(a,b) > $$ So we need: 1. $f_x$, $f_y$ to exist. 2. $f_{xy}$ and $f_{yx}$ to exist and be continuous at $(1,1)$. --- ### Step 2: Compute First Partial Derivatives Let $$ f(x,y) = \frac{x^2 - y^2}{x^2 ______ _________ _________ ___ _______.
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