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- Apr 14, 2013

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Hey!!

We define the sequence of functions $f_n:[0,1]\rightarrow \mathbb{R}$ by $$f_{n+1}(x)=\begin{cases}0 & \text{ if } x\in \left[ 0, \frac{1}{2n+3}\right ]\\ |2(n+1)x-1| & \text{ if } x\in \left [\frac{1}{2n+3}, \frac{1}{2n+1}\right ] \\ f_n(x) & \text{ if } x\in \left [\frac{1}{2n+1}, 1\right ] \end{cases}$$ where $f_1$ is given by $$f_1(x)=\begin{cases}0 & \text{ if } x\in \left [0, \frac{1}{3}\right ]\\ |2x-1| & \text{ if } x\in \left [\frac{1}{3}, 1\right ]\end{cases}$$

(a) Draw the graphs of the functions $f_n$.

(b) Show that the sequence of functions converges uniformly to a continuous function $f:[0,1]\rightarrow \mathbb{R}$.

(c) Calculate the lengths of the graphs of the functions $f_n$.

(d) Show that the graph of $f$ is not a rectifiable curve.

Could you give me a hint for (a) ? How do we draw these graphs?

At (b) we first show that $f_n$ converges pointwise to a function $f$, right? Taking the limit $n\rightarrow \infty$ then do we look at the intervals that we get at the definition of $f_{n+1}$ ?

We define the sequence of functions $f_n:[0,1]\rightarrow \mathbb{R}$ by $$f_{n+1}(x)=\begin{cases}0 & \text{ if } x\in \left[ 0, \frac{1}{2n+3}\right ]\\ |2(n+1)x-1| & \text{ if } x\in \left [\frac{1}{2n+3}, \frac{1}{2n+1}\right ] \\ f_n(x) & \text{ if } x\in \left [\frac{1}{2n+1}, 1\right ] \end{cases}$$ where $f_1$ is given by $$f_1(x)=\begin{cases}0 & \text{ if } x\in \left [0, \frac{1}{3}\right ]\\ |2x-1| & \text{ if } x\in \left [\frac{1}{3}, 1\right ]\end{cases}$$

(a) Draw the graphs of the functions $f_n$.

(b) Show that the sequence of functions converges uniformly to a continuous function $f:[0,1]\rightarrow \mathbb{R}$.

(c) Calculate the lengths of the graphs of the functions $f_n$.

(d) Show that the graph of $f$ is not a rectifiable curve.

Could you give me a hint for (a) ? How do we draw these graphs?

At (b) we first show that $f_n$ converges pointwise to a function $f$, right? Taking the limit $n\rightarrow \infty$ then do we look at the intervals that we get at the definition of $f_{n+1}$ ?

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