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If f is a continuous, positive function defined on the interval (0, 1] such that limx→0+...

If f is a continuous, positive function defined on the interval (0, 1] such that limx→0+ = ∞ we have seen how to make sense of the area of the infinite region bounded by the graph of f, the x-axis and the vertical lines x = 0 and x = 1 with the definition of the improper integral.

Consider the function f(x) = x sin(1/x) defined on (0, 1] and note that f is not defined at 0.

• Would it make sense to argue similarly and define the length of the infinite curve y = f(x) on the interval (0, 1]?

• If yes, what would be a reasonable suggestion for defining such a notion?

• For bonus marks find a differentiable (or, at least, continuous) function f such that limx→0+ f(x) = 0 and the graph of f on [0, 1] has infinite length.

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