Left and right continuity on unit square

Question

Let $T$ be function from $[0,1] \times [0,1]$ to $[0,1]$ satisfying

1. $T(T(x,y),z) = T(x,T(y,z))$
2. $T(x,y) = T(y,x)$
3. $T(t,x)$ and $T(x,t)$ is increasing on t
4. $T(1,x) = T(x,1) = x$

Function $T$ is said to be left-continuous if for every $x,y \in (0,1]$ $$T(x,y) = \sup\{T(u,v) \mid 0 < u < x, 0 < v < y\}$$ and $T$ is said to be right-continuous if for every $x,y \in [0,1)$ $$T(x,y) = \inf\{T(u,v) \mid x < u < 1, y < v < 1\}$$.

1. If $T$ is continuous on the usual Euclidean metric space, is $T$ both left-continuous and right-continuous?
2. If $T$ is both left and right continuous, if $T$ continuous on the usual Euclidean metric space?

I somehow think 1 true. But for 2, I can only prove that $T$ is continuous on $(0,1) \times (0,1)$. Are there any counter examples or a counter-example for my hypothesis?

Note : Function satisfying 4 conditions above is called a triangular norm