Find an example of a function $ f: \mathbb{R^2} \to \mathbb{R} $ so that $f$ is not continuous at $(0,0) \in \mathbb{R}$, but $ \lim_{t \to 0} f(tx) = 0$ where $tx= (tx_1, tx_2)$ (so $t \in \mathbb{R} $ and $x \in \mathbb{R^2}$). How can I tell my example works? So I chose $f: \mathbb{R}^2 \to \mathbb{R}$, $f(x,y)= \frac{x^2y}{x^2+y^2}$ since it is not continuous at $(0,0)$, but then I don't know how to prove that $ \lim_{t \to 0} f(tx) = 0$.
Finding non-continuous function
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real-analysis
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0You must mean $f:\mathbb R^2\to \mathbb R$, not how you wrote it. Can you find common factors of $t$ when you write explicitly what $f(tx)$ is for your example? – 2012-11-28
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1Another example: $f(x_1,x_2)=1$ if $0\neq x_2=x_1^2$, $f(x_1,x_2)=0$ otherwise. – 2012-11-28
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0$f(tx)= \frac{t^2(x_1^2x_2^2)}{x_1^2+x_2^2}$ – 2012-11-28
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0Luz: That isn't correct. Did you use $y^2$ in the numerator instead of $y$? – 2012-11-28