In this short note, Monte Carlo tests of goodness of fit for data of the form $X(t), t\in I$ are considered, that reject the null hypothesis if $X(t)$ leaves an acceptance region bounded by an upper and lower curve for some $t$ in $I$. A construction of the acceptance region is proposed that complies to a given target level of rejection, and yields exact $p$-values. The construction is based on pointwise quantiles, estimated from simulated realizations of $X(t)$ under the null hypothesis.

Keywords: Cdf transform, functional data, multiple testing, multivariate data, simulation based tests.