The discontinuous paths do 'disappear' when you take the continuum limit. They contribute nothing to the integral in the end. In the Euclidean picture, they are suppressed by the kinetic term in $e^{-S(\phi)}$, which looks like $\sum_t \frac{(\phi(t+a) - \phi(t))^2}{a}$.

The measure you define by taking this limit is known as Wiener measure. If you're being particularly fussy, Wiener measure is defined on distributions, but has support only on the subset of distributions which are represented by continuous functions.

One of the little details that makes QFT more difficult than QM is that the fluctuations of fields are not generally suppressed in the continuum limit. In scalar field theory in 4d, the kinetic terms looks like $\sum_x \sum_\mu a^2 (\phi(x + a e^\mu) - \phi(x))^2$, so you can get exponent 2 power law singularities in the correlation function of two field values.

This post imported from StackExchange Physics at 2014-05-04 11:13 (UCT), posted by SE-user user1504