Real Analysis/Arc Length
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Suppose we have a parametric curve in three dimensions, f(t) = (x1(t),x2(t),x3(t)). Of course, it would be required that all three functions be continuous. This essentially defines a curve, since it is a continuous image of the real numbers onto the real 3-space.
Now, we can define the arc length of this curve over an interval. Say the interval is [a,b]. Now divide [a,b] into partitions, a = a0 < a1 < a2 < a3 < ... < an = b, and call this partition P. Take the sum of the distances | f(an) − f(an − 1) | , to get 
, and call this sum L(P). Now, take the supremum of the lengths, 
is a partition}. If this number is finite, we call it a rectifiable curve.
Now we establish a sufficient and necessary condition for a curve in 3-space to be rectifiable (note: this can easily be extended to an n-space through an analogous argument).
Theorem:
A continuous curve in three dimensions is rectifiable if and only if all of its component functions are functions of bounded variation.
Proof:
Theorem:
If a curve f(x) in 3-space is continuously differentiable in all 3 components, then it is rectifiable and the length from f(a) to f(b) is 
.
Proof:
