motivated by email from Gary L.
I'm looking for a particular kind of function ... something continuous everywhere, but differentiable nowhere.
We start with g(x) which looks like Figure 1A.
In fact, g(x) = x for 1/2 < x < 1/2 and it's extended periodically.
Note that g(x) = 0 when x = n, an integer ... and g(x) = 1/2 when x = (2n+1)/2.
Then we construct g(4x) / 4 which looks like Figure 1B.
>Huh?
Since g(x) = 0 when x = n, then g(4x) = 0 when 4x = n or x = 1/4, 2/4, 3/4 etc. etc.
Also, g(x) = 1/2 when x = (2n+1)/2 so g(4x) = 1/2 when 4x = (2n +1)/2 or x = 1/8, 3/8, 5/8 etc. etc.
In between these values, g(4x) has that sawtooth character.
By dividing by 4 we reduce the size by a factor of 4 ... giving Figure 1B.
Now we construct g(16x) / 16 which is zero when 16x = n or x = 1/16, 2/16 and ...
>Yeah, I get it ... but then what?
 Figure 1A
Figure 1B

Then, having constructed g(x) and g(4x)/4 and g(16x)/16 and, for every positive integer n, g(4^{n}x)/4^{n}, we add them all up.
This gives F(x) = _{Σ} g(4^{n}x)/4^{n}, the sum going from n = 0 to n = ∞.
>I assume "F" stands for Funny Function, right?
You got it!
>So what does it look like?
Uh ... well ... I can't draw the whole infinite sum, but I can give you a few partial sums.
>And it's continous but nowhere differentiable?
Yes.
Let f_{n}(x) = g(4^{n}x)/4^{n} so F(x) = f_{0}(x) + f_{1}(x) + f_{2}(x) + ...
Then f_{n}(x) has a "corner" (hence isn't differentiable) whenever it has a zero value
... and that occurs 4^{n} times in 0 < x ≤ 1, namely where 4^{n}x = 1, 2, 3 ... 4^{n} or x = 1/4^{n}, 2/4^{n}, 3/4^{n} ... 1.
Wherever any f_{n}(x) fails to be differentiable, our Funny Function F(x) won't be differentiable.
Clearly, the sum of all f_{n}(x) will then fail to be differentiable at infinitely many places in 0 < x ≤ 1 and, by periodic extension, to all x.
One of these places will lie in every interval (x_{0}  ε, x_{0}  ε) regardless of the values of x_{0} and ε > 0.
Since the set of points of nondifferentiabiliy is everywhere dense on the xaxis, our Funny Function will be differentiable nowhere.
>And with so many corners, it isn't continuous, right?
No, no ... it's the uniformly convergent sum of continuous functions, is so it's continuous.
>Yeah, so where's the graph of F(x)?
The functions g(x) and g(4x) and g(16x) look like this:
If we reduce the size of each (as noted above) we'd get this:
>I don't see all those corners ...
Of course not. The maximum value of f_{2}(x) is just 1/32 = 0.03125 and that's pretty small, eh?
>And this Funny Function is your invention?
Are you kidding? There are lots of such Funny Functions: everywhere continous but nowhere differentiable, everywhere differentiable but nowhere monotone, everywhere ...
>Do they have names?
Sure, like Kopcke functions.
Note that all of our f_{n}(x) functions are periodic and have a Fourier Series expansion ... and the
Weierstrass Function is such an animal.
>And if I search for "Funny Functions" will I get some more?
I have no idea, but try it !
>Aha! Even the name "Funny Function" ain't yours !
Of course not.
>So where did this probem come from?
There was a question concerning a function satisfying:
f(1) = 5, f(1) = 2, f(5) = 5 and f(8) = 1.
The question concerned the existence of a point of inflection.
An example is shown in Figure 2, where two parabolas are joined at x = 5 where there is a point of inflection.
Now the question:
Can one guarantee a point of inflection ... if the function is continuous?
>I'd say no.
And you'd be right.
 Figure 1A

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