Why are distributions defined to be continuous wrt. the test functions?

Question

Test functions are equipped with a topology of uniform convergence of all derivatives on all compact subsets. Distributions are then defined as linear functionals that are continuous wrt. that topology. Distributions are themself also equipped with a topology, namely the weak* topology which is the same as product topology. Any linear operator on test functions can be adjointed to a linear operator of distributions, and it will automatically be continuous wrt. the weak* topology. For example differentiation. The requirement of continuity wrt. test functions is not used when performing such operations on distributions. What is its purpose?