For a function $f: \mathbb{R}^n \to \mathbb{R}^m$, I know that continuity and sequential continuity are equivalent.
Sequential continuity of $f$ at a point $x \in \mathbb{R}^n$ means for any sequence in the domain that converges to $x$, the function values on the sequence also converge to $f(x)$. I feel this definition difficult to apply even when the domain is $\mathbb{R}$ or its subset, because how can one possibly consider all sequences that converge to $x$? There are many cases that a sequence can converge to $x$.
On the other hand, continuity of $f$ at a point $x \in \mathbb{R}^n$ means for any neighbourhood of (or just open ball centered at) $f(x)$, there exists a neighbourhood of (or just open ball centered at) $x$, whose image under $f$ is contained in the one of $f(x)$. I feel this definition is easier to apply, but I don't know how to explain why.
So is my understanding of proving sequential continuity more difficult than proving continuity right?
If needed, here is an example: prove $f([x,y])=x-\sqrt{y}$ is continuous at $(1,1)$.
Thanks and regards!