Continuous everywhere but differentiable nowhere

This post is again on limits.

In my class, somebody asked our Maths teacher a question about a function that was continuous but not differentiable.

But first, a little backstory.

Our teacher taught us about continuity and differentiability in the following manner:

At a point,

1. the limit exists.
2. the function is continuous.
3. the function is differentiable.

If #1 is true, it’s akin to the function passing its matriculation exam (the 10th grade). If #2 is true, it can be thought of as the function completing its undergraduate course at IIT (Indian Institute of Technology). If #3 is true, it corresponds to the function completing its graduate course at IIT.

If #1 is false, there is no point in talking about #2 and #3. Similarly, if #2 is false, there is no point in talking about #3.

By the same logic, if #2 is true, there is no point in inquiring whether #1 is true or false, and if #3 is true, finding out whether #1 and #2 are true makes no sense.

In other words:

Statement 1: A function is continuous at a point only when the limit exists at that point.

Statement 2: And, a function is differentiable at a point only when it is continuous at that point.

Expressing the above two statements to read concisely would yield the following:

Corollary 1: The limit of a continuous function always exists.

Corollary 2: A differentiable function is always continuous.

Using the simple logic of what is called mathematical reasoning, we can deduce the following two statements:

Statement 3: A continuous function need not be differentiable.

Statement 4: A function whose limit always exists need not be continuous.

This post has to do with statement 3 above. The only example of a continuous function that was not differentiable was a function whose domain (and, by the definition of a function, range) was a set of isolated points.

But there are functions which are continuous everywhere, but differentiable nowhere. These functions, when “zoomed in” do not start to resemble a straight line. In other words, no matter how minute an interval of domain you take, these functions will never be monotonous in any such interval.

These functions were the precursors to what are called fractals. It’s difficult to explain, but typing in “define fractal” on Google yields the following definition of sorts:

fractal: a curve or geometric figure, each part of which has the same statistical character as the whole. Fractals are useful in modeling structures (such as eroded coastlines or snowflakes) in which similar patterns recur at progressively smaller scales, and in describing partly random or chaotic phenomena such as crystal growth, fluid turbulence, and galaxy formation.

So one of the functions which are continuous everywhere but differentiable nowhere is the Weierstrass function, which was published back in 1872. It is defined as ƒ(x) = ∑_{n=0 to ∞} (aⁿ cos(bⁿ πx)), where 0 < a < 11, ‘b’ is a positive odd integer, and ab > 1 + 3π/2.

(Oh yeah!)