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## Main Question or Discussion Point

A parametric equation, say

1. its derivative is continuous, and

2. its derivative does not equal zero for all t in the domain of r.

Now that sounds simple enough. Now lets say we have the

then

Now, without reverting to MatLab or Maple to view the graph, how do we mathematically explain that

Do I just state that is is/isn't -by inspection, or ...?

**, is smoothly parametrized if:***r(t)*1. its derivative is continuous, and

2. its derivative does not equal zero for all t in the domain of r.

Now that sounds simple enough. Now lets say we have the

*tractrix*:**, ...***r(t) = (t-tanht)i + sechtj*then

**, right?***r'(t) = [ 1/(1+x^2) ]i + [ tantsect ]j*Now, without reverting to MatLab or Maple to view the graph, how do we mathematically explain that

**is continuous (or not)?***r'(t)*Do I just state that is is/isn't -by inspection, or ...?