Most people (probably outside mathematics) think that continuous functions are those that can be drawn without lifting the pencil from the paper.

But this is actually false. For example it is well known that the set of continuous nowhere differentiable functions is prevalent (its complement has, in some sense, of infinite-dimensional-measure equals zero), residual (its complement is of first Baire category), it is spaceable (it contains, except for the null-function, a closed infinite dimensional vector space) and many more properties that suggest to us it is a VERY BIG set.

And what have this to do with being drawable? Well, any of these functions cannot be drawn with a pencil: If you draw some (enough) small line, you do it upwards, downwards or horizontally, so it must have positive, negative or null derivative (on its whole domain, say a small interval).... BUT the functions is NOT derivable.

I used to call this, "The great lie of continuity".