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Define $f$ on $\mathbb{R}^{2}$ by $f(x,y) = x^{3}\sin(1/x) + y^{2}$ if $x \neq 0$ and $f(0,y) = y^{2}$. I've got $D_{1} f(x,y) = 3x^{2}\sin(1/x) - x\cos(1/x)$ if $x \neq 0$ and $D_{1} f(0,y) =0$. (I've used the definition, not just a formula.)

The textbook says 'Show that $(\partial / \partial x) f$, i.e., the $x$-partial derivative of $f$, is discontinuous at $(0,0)$' but how come I get it is cts at $(0,0)$?

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