Proving differentiability

Determine whether or not $\displaystyle f(x,y)=x^{2}+3xy$ is differentiable.

Now, from what I understand, in order to show that a function of two variables is differentiable at some point $\displaystyle (a,b)$, I have to show that $\displaystyle \frac{\partial f}{\partial x}$ and $\displaystyle \frac{\partial f}{\partial y}$ are continuous at $\displaystyle (a,b)$ and that $\displaystyle f(x,y)$ itself is continuous at $\displaystyle (a,b)$. I know that the partials are continuous for all $\displaystyle (x,y)$, but how do I show that $\displaystyle f(x,y)$ is continuous? (The problem does not ask if the function is differentiable at a specific point $\displaystyle (a,b)$.) Thanks in advance.

Same question with the equation $\displaystyle f(x,y)=xy^{2}$.