Boundedness of continuity on higher dimension So I studied that any continuous functions $f:[a, b]\to \mathbb{R}$ then f is bounded, is this generally true for higher dimensional functions, say $f:[a, b] \times[a, b]\to \mathbb{R}$?

Sure because the domain of $f$ is compact and $f$ is continuos. This means that $|\cdot|\circ f:[0,1]\times[0,1]\to \mathbb{R}$ is continuos. 
Every continuos function maps compact sets to compact sets, so 
$(|\cdot |\circ f)([0,1]\times[0,1])\subseteq \mathbb{R}$ is compact
but each compact set of $\mathbb{R}$ is closed and limited, so 
$|f(x,y)|$ is limited in $\mathbb{R}$.
