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

This gives F(x) = _Σ g(4ⁿx)/4ⁿ, 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(4ⁿx)/4ⁿ so F(x) = f₀(x) + f₁(x) + f₂(x) + ...
Then f_n(x) has a "corner" (hence isn't differentiable) whenever it has a zero value ... and that occurs 4ⁿ times in 0 < x ≤ 1, namely where 4ⁿx = 1, 2, 3 ... 4ⁿ or x = 1/4ⁿ, 2/4ⁿ, 3/4ⁿ ... 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 (x₀ - ε, x₀ - ε) regardless of the values of x₀ and ε > 0.
Since the set of points of non-differentiabiliy is everywhere dense on the x-axis, 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 f₂(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