# Continuity properties of sample functions

## Abstract

In what follows we shall give further theorems concerning almost all (3) sample functions. It is important to distinguish between two kinds of assertions: (i) an assertion about x (*t,* ω) for a. a.ω at a *fixed t* which is the same for allω; (ii) an assertion about *x*(•,ω ) for a. a.ω at a *generic* regarded as the running adscissa in the sample graph (*t,x*(*t*, ω) ),t∉*T*,for each ω. thus the exceptional null set may depend on *t* in case (i) but not in case(ii) in Naturally (i) is a special case of (ii). As anc an example, the second assertion in Theorem 5.6 may be stated as fol- lows: for a fixed *t* and stable *i, t∉ S*_{ i } (w) implies that *t* is in an (open) *i*-interval of *x*(•, ω) for a.a. w. This is an assertion of the first kind; the corresponding assertion of the second kind is false at a generic *t* which is an endpoint of an i-interval. Similarly, Theorem 2 below is for an assertion of the first kind while the corresponding assertion of the second kind will be given in Theorem *7*.4.

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