the continuous, nowhere-differentiable paths

Zoom in on a smooth curve and it eventually looks like a straight line — that is what 'has a derivative' means: a well-defined slope, a velocity, at every point. Now zoom in on a Brownian path. It never straightens out. Magnify any tiny piece and you see the same crinkly chaos as the whole; there is no slope to read off, no instantaneous velocity, anywhere. A Brownian path is continuous (it has no gaps or jumps — you can draw it without lifting your pen) yet differentiable at no point at all. It is the textbook monster: an unbroken curve with no direction.

Why does smoothness fail? Look at the difference quotient that defines the derivative at time t: [B(t + h) - B(t)] / h. The numerator is a Normal(0, h) random variable, so its typical size is sqrt(h). Dividing by h gives a typical size of sqrt(h)/h = 1/sqrt(h). As h shrinks toward 0, that blows up to infinity — the would-be slope has no finite limit, it oscillates wildly without settling. This happens at every t simultaneously (with probability one), which is why the path is nowhere differentiable. The intuitive cause is that the increments are independent: the next instant carries fresh randomness completely unrelated to the current direction, so the path can never commit to a tangent.

This is not a pathological curiosity to be waved away — it is the structural fact that reshapes everything downstream. Because there is no velocity dB/dt, you cannot write an ordinary differential equation driven by Brownian motion; the symbol 'dB' has to be reinterpreted through the Ito integral. And because the path is too rough to have a slope but just rough enough to accumulate a finite 'quadratic variation', the ordinary chain rule fails and Ito's lemma takes its place. The roughness is the source of the entire stochastic calculus, not an embarrassing footnote.