EDIT: To make sense, the function in the question must be some $f:[0,1]\times [0,1]\rightarrow \mathbb{R}$. Then, what I want to show is that $f$ is continuous with respect to its second argument. This is why I choose $\lambda$ equal to the sup (or max) of $|f(x)-f(x_0)|$, which does not depend on the first argument but it does depend on $x$ and $x_0$
I am not sure about the use of \lambda$ when negating continuity. Fix the domain of $f$. Suppose that $f$ is not continuous on $[0,1]$. Then, negating the def of continuity we get: $\exists \,x_0\in[0,1]$ $\exists \, \epsilon>0$ such that $\forall \lambda>0$ $\exists x\in [0,1]$ such that $|x-x_0|<\lambda \text{ and } |f(x)-f(x_0)|\geq \epsilon$. My understanding is that this negation should work for every $\lambda>0$ which allows me, for example, to choose $\lambda=\sup_{t}|f(x)-f(x_0)|$. My doubts is that this particular $\lambda$ depends on $x$ as well as on $x_0$ and I'm not sure this is correct.
Any help is greatly appreciated it!