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# Topology

Explore geometric properties and spatial relations that are unaffected by continuous deformations, like stretching and bending. The next time you scan a bar code on a can of soda, thank a topologist.

# Deep Dive: Pointwise and Uniform Convergence 1

A sequence of functions $$f_n(x)$$ is drawn in stages. Each stage in the sequence consists of a connected path of semicircles:

$$f_1(x)$$ consists of 1 semicircle of radius 1,

$$f_2(x)$$consists of 2 semicircles of radius $$\frac{1}{2},$$

$$f_3(x)$$consists of 4 semicircles of radius $$\frac{1}{4},$$

and so on.

As $$n$$ increases and this process continues indefinitely, what happens to the lengths of $$f_n(x)$$ over the interval $$[0,2]?$$

True or False?

Let $$f_n(x)$$ be the function on $$[0,2]$$ whose graph is the Stage $$n$$ semicircles. Then $$f_n\left(\frac{1}{3}\right) \rightarrow 0,$$ i.e.,

$\lim_{n \to \infty} f_n\left(\frac{1}{3}\right) = 0.$

True or False?

Let $$f_n(x)$$ be the function on $$[0,2]$$ whose graph is the Stage $$n$$ semicircles. Then for all $$x$$ in $$[0,2],$$ $$f_n\left(x\right) \rightarrow 0,$$ i.e.,

$\lim_{n \to \infty} f_n\left(x\right) = 0.$

Note. Another way of phrasing the question is: "Does the sequence $$\{f_n(x)\}$$ converge pointwise to $$f(x) = 0$$ on the interval $$[0,2]?$$

True or False?

Let $$f_n(x)$$ be the function on $$[0,2]$$ whose graph is the Stage $$n$$ semicircles. Then for all $$x$$ in $$[0,2]$$ where they are defined, the derivatives $$(f_n)'\left(x\right) \rightarrow 0,$$ i.e.,

$\lim_{n \to \infty} (f_n)'\left(x\right) = 0.$

Note. 0 is the derivative of the function $$f(x) = 0$$ that the functions $$f_n$$ approach.

True or False?

Suppose $$\{f_n\}$$ and $$f$$ are functions on an interval $$I,$$ and $$L(g)$$ represents the length of the curve $$g$$ on $$I.$$

If $$f_n(x) \to f(x)$$ for all $$x \in I,$$ then $$L(f_n) \to L(f).$$

Alternate phrasings:

If $$\lim_{n \to \infty} f_n(x) = f(x)$$ for all $$x \in I,$$ then $$\lim_{n \to \infty}L(f_n) = L(f).$$

If the sequence of curves approach $$f$$ at each point, then the lengths of the curves approach the length of $$f.$$

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