If we have injective continuously differentiable function $f$, and if we define continuously differentiable function $g(x,y)=(f(x),y)$, then does it follow that $g$ is injective?
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2026-04-19 12:04:49.1776600289
If component functions are injective then is the function injective?
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Recall that for $g$ to be injective means that $g(x_0,y_0) = g(x_1,y_1)$ implies that $(x_0,y_0) = (x_1,y_1)$. Start by assuming the antecedent and try to prove the consequent. (Note that you don't need to use the fact that either $f$ or $g$ is continuously differentiable.)