Let $f:\mathbb{R_{\geq 0}}\times \mathbb{R}^n\to\mathbb{R^n}$ be an arbitrary function, e.g., with $n=1$, $f(t,x) = t^2+x.$
What is the difference among the following expressions:
"$f$ is continuous in $\mathbb{R_{\geq 0}}\times \mathbb{R}^n$"
"$f(t, \cdot)$ is continuous for each fixed $t$"
"$f$ is continuous in its second argument $x$"
Of course, the literal meaning of both expressions is obvious. However, my questions is how are they (or aren't they) related?
To further Git Gud's comment.
The first is often called joint continuity. The second is continuity in the first component.
Even if both $f(\cdot, t)$ and $f(x,\cdot)$ are continuous, joint continuity is not necessarily true, but joint continuity implies component wise continuity.
See here for an example:
Does factor-wise continuity imply continuity?
The answer for this post illustrates this point.
Edit: your second and third line means the same thing.
