I was going through the notes of my professor. There it is mentioned that ''We say that a stochastic process ($X_{t}$) is continuous (or, equivalently, has continuous paths) if, for almost all ω ∈ Ω, the function t → $X_t(ω)$ is continuous.''
Suppose I have a continuous time stochastic process, ($X_{t}$). Suppose all of my random variables are iid random variables such that $X_{t}(\omega~')=2$ and $X_{t}(\omega~'')=3$ $\forall t$ where $\Omega=\{\omega~',\omega~''\}$ i.e. the sample space contains just these 2 outcomes. Now, suppose that $\omega~'$ appears just once at time k and after that only $\omega~''$ appears. So, there will be a discontinuity at time k. Isn't this violating the definition of continuous stochastic process or is it that I have to keep $\omega$ constant throught out the process ?
Also, is $\omega$ in the definition of continuous stochastic process the outcome at any point of time or is it the string of the outcomes that occurs till time infinity?
The set $\Omega$ is the sample space, which can be any set you want. Since $\Omega$ is a set, though, it doesn't make sense to say that $\omega \in \Omega$ "appears just once at time $k$". Elements of sets don't appear at any time; they just are in the set.
A random variable $Z$ is a function mapping $\Omega$ to $\mathbb{R}$ (or another set, but let's just use $\mathbb{R}$ for now), i.e. $Z(\omega) \in \mathbb{R}$ for all $\omega \in \mathbb{R}$. A stochastic process $X$ is a collection of random variables $(X_t)_{t \in \mathcal T}$ where $\mathcal T$ is some index set, e.g. $\mathcal T = [0,\infty)$. We say $X$ is a continuous stochastic process if, for almost all $\omega \in \Omega$, we have the function $t \mapsto X_t(\omega)$ is continuous. Note that in this definition, we only care about $t \mapsto X_t(\omega)$ while $\omega$ stays fixed.
Incidentally, as a side note, the example you provided where $X_t(\omega') =2$ and $X_t(\omega'') = 3$ for all $t$ isn't really an iid sequence because $X_t(\omega) = X_s(\omega)$ for all $t,s$, and $\omega \in \Omega$.