Intuitive meaning of smooth curve A curve ,let's say $(x(t),y(t))$ is said to be smooth if $x'(t)$ and $y'(t)$ both exist and are continuous.(Am I not right?)
A function differentiable at a point intuitively means that its graph on coordinate plane has a unique tangent(no corner) at that point.
So what does a smooth curve mean intuitively ? 
Thanks in advance!
 A: Here is the best way I can think of to explain smoothness intuitively.

Pardon the crappy picture.
Smoothness is all about making sure you're ending up at the same place coming from the left or the right.
Pretend you're in a car driving really FAST on this hill.
Suppose you're driving from left to right. The transition from red to blue will be no big deal. The car will maintain contact with the road. Similarly, if you're driving right to left from blue to red, the car will transition that. This is smooth.
Now suppose the car has reached that jagged peak. If it doesn't look jagged enough to you, pretend it is. I'm no artist. Because the car is going really fast, it wants to fly off, following that blue trajectory. The car will NOT maintain contact with the road. Similarly, from right to left, the car wants to follow that green trajectory. Thus, where the fast car wants to go when coming from the left does not agree with where the fast car wants to when coming from the right. This is NOT smooth.
Hope this makes some sense.
A: In single variable calculus (at Georgetown), I was taught that a smooth curve is one which has all of the following:
a) you can draw it without lifting your pen of the paper
b) It has no "sharp edges" (for instance the function |x| at 0 is "sharp"). Note that the meaning of smooth is imprecise, for instance I would call this function smooth, but its not differentiable at a because it has vertical tangent lines.  
This site has some good examples: http://www.zweigmedia.com/RealWorld/calctopic1/contanddiffb.html
In two dimensions its a bit harder to visualize, since you need a third dimension to plot the function. Consider some of "Quadric Surfaces" seen here for examples of "smooth functions" with 2 dimensional domains:
http://tutorial.math.lamar.edu/Classes/CalcIII/QuadricSurfaces.aspx
As to the notion of continuity, in Real Analysis you'll see 2 kinds of continuity, pointwise and uniform, and you'll see in topology that these come from there. And down the rabbit hole you go...
