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I'm reading a paper where the authors describe a function $f(x, y)$ that is 'continuous in its first argument'. Specifically, $x \in [0, 1]$ while $y \in \{0, 1\}$. I can't find the definition for a function that is continuous in its first argument, though I expect it is obvious. Can anyone help?

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  • $\begingroup$ It means that the function $\phi_y(x) = f(x,y)$ is continuous (for fixed $y$). $\endgroup$ – copper.hat May 28 '20 at 17:18
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All it means is the following. For fixed $y$, define $g_y(x):=f(x,y)$. That $g_y$ is a continuous function.

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In your case it will mean that for any sequence $x_n$, which converges to some point $x$, you will have that $f(x_n, y) $ converges to $f(x, y)$, but not necessarily the same holds for a sequence $y_n$ and $f(x, y_n) $. Is that what you were looking for?

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