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I googled and came to know that there are many continuous functions which cannot be drawn by hand, like Cantor, Weierstrass functions etc.

Now this question was asked in a college admission interview. We had to explain our answer. So what and how to approach the answer?!

Ranojoy Dutta
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    there are many continuous functions that you cannot draw. I suppose maybe the rigorous sense of the word is "path connected" in which case, you an see the topologist's sine curve, which is connected but not path-connected – Andres Mejia May 29 '16 at 06:16
  • @AndresMejia I considered the path connectedness, but, probably nowhere differentiable continuity is much more appropriate for pre-college admission, although still somewhat unexpected. – Emre May 29 '16 at 06:25

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Disclaimer: None of the following arguments are rigorous, just to give an intuition of a nowhere differentiable continuous function.

The continuous functions are, in some sense, those functions that have connected graphs.

First of all, one needs to realize, there are non-smooth(in the sense that the graph has spikes) continuous functions. For example, $|x|$ has a spike at $0$.

Moreover, you can see a zigzag connecting the point $(2n-1,1)$ to $(2n,0)$ and point $(2n,0)$ to $(2n+1,1)$ for every $n\in\mathbb{Z}$ also defines a continuous function. However, this function is still drawable, as the points that you need to change direction are spaced enough.

Nextly, decrease the height of this zigzag, and make the spikes more frequent, i.e. connect the points of the form, $(n,0)$ and $(n+1/2,1/2)$.

Keep squeezing the zigzag and making the space narrower. Then, add all these zigzags to get a new function, which would be defined finitely as the sum of $2^{-n}$ is finite. Your final function will have spikes with frequency $2^{-k}$ for every $k$, as $2^{-k}\to0$, the points where you need to change the direction will be too often for you to change the direction.

Emre
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