Very interesting question – it took me a minute to figure this one out. Let's look at a simpler case of 2 variables. Let x=t3 and y=t2. Notice that the derivatives of both functions (with respect to t, don't forget the principles you are actually applying) are indeed smooth curves as you pointed out.
However, to find the derivative with respect to x we can use the chain rule, or alternatively solve for y in terms of x and then differentiate w.r.t. x. The first way is easier and yields:
dy/dx=2/(3t), which is indeed discontinuous at t=0.
