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MATH114 Integration and Differentiation

Problem Solving Class 3

[2.5ex]

Step 1

Name a function $f_{0}:[-1,1]\rightarrow\mathbb{R}$ which is everywhere continuous but not everywhere differentiable. Do you think a function could exist which is everywhere continuous but nowhere differentiable?
Step 2
Write down a function $f_{1}:[-1,1]\rightarrow\mathbb{R}$ with the following properties:
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  $f_{1}$ is continuous,
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  $f_{1}$ is not differentiable at $-\frac{1}{2},0,\frac{1}{2}$ ,
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  $|f_{1}^{\prime}(x)|=\frac{1}{2}$ at all points $x$ where $f_{1}$ is differentiable.
Start graphically using step 1 and remembering how to express continuity and differentiability graphically, then turn it into a formula.
Step 3
Generalising step 2, draw the graph of a function $f_{n}:[-1,1]\rightarrow\mathbb{R}$ with the following properties:
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  $f_{n}$ is continuous,
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  $f_{n}$ is not differentiable at $\{\frac{k}{2^{n}}:k=-2^{n}+1,\ldots,2^{n}-2,2^{n}-1\}$ ,
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  $|f_{n}^{\prime}(x)|=\frac{1}{2^{n}}$ at all points $x$ where $f_{n}$ is differentiable.
Do that for a few $n$ of your choice and notice that $n=1$ gives back the function in step 2. It would be helpful to construct the family $(f_{n})_{n\in\mathbb{N}}$ in a somehow consistent recursive way.
Step 4

Translate the graphs of $f_{n}$ into a proper definition of $f_{n}$ .
Step 5

Find a function $F:[-1,1]\rightarrow\mathbb{R}$ which is nowhere differentiable, i.e., where the points of non-differentiability lie infinitely close to one another. Justify briefly but do not try to prove all the details.
Step 6

Can you draw the graph of the function $F$ in step 6? What does that tell you about “graphical proofs” for functions?