# What is it for a function with two argument places to be continuous in its first argument?

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?

• It means that the function $\phi_y(x) = f(x,y)$ is continuous (for fixed $y$). – copper.hat May 28 '20 at 17:18

All it means is the following. For fixed $$y$$, define $$g_y(x):=f(x,y)$$. That $$g_y$$ is a continuous function.
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?