[FOM] A question concerning continuous functions

[Dana Scott]

It seems to me that the problem with the "chalkboard definition" is
that it is circular: continuous motion (in 2D) is used to define
continuous graphs (for 1D functions).

But the idea of the characterization is really "no jumps" -- or at
least that is what I always taught.  The epsilon-delta definition is
quite clear on this: to every horizontal strip around a point on a
graph of a function to be tested for continuity, you can find a
vertical strip such that the graph stays completely inside the
rectangle (= the intersection of the two strips).  

In other words, no jumps, no lifting the chalk (pen/pencil).  As you
draw, you have to stay close to previous points.  Is there anything
deeper here?  Does the definition not make an intuitive idea rigorous?