My Journey into Constructions of Pathological Functions

When I first learned about continuous but nowhere differentiable functions, I was fascinated. The idea that you could draw a function without lifting your pencil, yet never find a tangent line anywhere, felt like a paradox. How could such functions exist? And more importantly — could I discover one myself, using only the tools I knew?

This question eventually led me to write my paper Functional Equations and Combinatorial Constructions of Pathological Functions in Real Analysis. But the road there was more like a chain of small discoveries than a single big idea.      check out the full paper

---

The Starting Point: Weierstrass and Takagi

My journey began with the Weierstrass function:

$$W(x) = \sum_{n=0}^\infty a^n \cos(b^n \pi x).$$

It’s elegant, but all the proofs I could find that $W$ is nowhere differentiable used harmonic analysis. Fourier series, lacunarity arguments, deep inequalities. I thought: there has to be a simpler way.

Then I found the Takagi function:

$$T(x) = \sum_{n=0}^\infty 2^{-n} \tau(2^n x), \quad \tau(x) = \text{dist}(x,\mathbb{Z}).$$

This was more combinatorial, more elementary. It felt like something I could play with directly. So I asked: what if I tweak the coefficients and scaling? What happens to differentiability?

---

Playing with Functional Equations

I noticed that Takagi’s function satisfies a functional equation. If you zoom in by a factor of 2 and scale down, you get the same function back. That suggested fixed-point methods.

So I tried writing something like:

$$(TF)(x) = g(x) + \alpha F(\beta x),$$

and wondered: what kind of functions arise as fixed points of this operator?

By applying the Banach fixed-point theorem, I realized I could generate functions of the form:

$$F(x) = \sum_{n=0}^\infty \alpha^n g(\beta^n x).$$

This was a big moment: I had a whole family of functions generalizing Takagi.

Now the question became: which of these are smooth, and which are pathological?

---

The Phase Transition

The key was analyzing difference quotients:

$$Q(x,h) = \frac{F(x+h) - F(x)}{h}.$$

For the generalized Takagi function

$$T_{\alpha,\beta}(x) = \sum_{n=0}^\infty \alpha^n \tau(\beta^n x),$$

I derived the identity:

$$Q(x,h) = \frac{\tau(x+h) - \tau(x)}{h} + \alpha\beta \, Q(\beta x, \beta h).$$

Iterating this relation was like peeling back layers of self-similarity. Eventually, I saw the pattern:

• If $\alpha\beta < 1$, the difference quotients remain controlled → Lipschitz continuity.

• If $\alpha\beta \geq 1$, the quotients blow up along some sequence → nowhere differentiability.

It felt like discovering a phase transition: just by crossing the threshold $\alpha\beta = 1$, everything changes.

---

Digits and Irregularity

But I wasn’t done. I kept thinking about whether I could build a pathological function directly from binary digits. After all, numbers in $[0,1]$ have expansions:

$$x = \sum_{n=1}^\infty d_n(x) 2^{-n}, \quad d_n(x) \in \{0,1\}.$$

What if I turned these digits into weights in a series?

So I tried:

$$f_p(x) = \sum_{n=1}^\infty \frac{d_n(x)}{n^p}, \quad p > 1.$$

It converges uniformly (since $\sum n^{-p} < \infty$), so $f_p$ is continuous.

But what about differentiability?

I realized that binary expansions let you flip digits locally. With a small perturbation $h$, you can change the $m$-th digit while keeping earlier digits fixed. That gave me a lower bound:

$$\Bigg|\frac{f_p(x+h) - f_p(x)}{h}\Bigg| \gtrsim \frac{2^m}{m^p}.$$

And since $2^m$ grows faster than $m^p$, the difference quotients diverge. So $f_p$ is nowhere differentiable.

It was so simple, yet so effective: pathology born from digit flips.

---

Putting It Together

At this point, I had two independent constructions:

1. Functional equations (self-similarity, scaling).

2. Combinatorics of digits (binary expansions).

Both led to continuous nowhere differentiable functions, but from very different angles. To me, that was the real beauty: pathology isn’t a single trick, it’s a recurring theme in analysis.

---

What I Learned

• Sometimes, deep theorems (like Hardy’s) can be approached with elementary tools if you find the right perspective.

• Self-similarity and digit expansions are incredibly powerful — they hide wildness inside simple formulas.

• Doing math isn’t about memorizing results. It’s about playing with definitions, tweaking them, and seeing where they lead.

When I started, I didn’t expect to rediscover known results, let alone prove new ones. But by following curiosity, I ended up with a framework that feels natural and elegant.

---

Closing Thoughts

Pathological functions remind us that analysis is not about smooth curves drawn with a steady hand. It’s about exploring the edge cases, the counterexamples, the surprises that force us to rethink what continuity and differentiability really mean.

For me, this journey wasn’t just about proving theorems — it was about discovering beauty in chaos, and realizing that mathematics rewards curiosity as much as rigor.

---