In Mathematical Finance, is it possible to have a claim that is continuous function of a factor model? Assume that the factor model $X_{t}$ is a Ornstein-Uhlenbeck process and the claim $\xi$ at maturity $T$ is a function of the factor model $\xi_{T}=f(X_{T})$. For $\xi_{t}=f(X_{t})$ at any time $t$, when can I assume that the function $f$ is continuous?
 A: You can regard both underylings and claims as subsequences and continous functions of subsequences of real numbers. If you really need to appeal to them there of course are standard theorems linking the discretised and continuous behaviour in $\mathbb{R}$. Even if measurement and payoff evaluation is necessarily discrete for practical purposes.
Possibly the first payoff you encounter in math finance: $(X_T-K)^+$ is an example of a function continuous in $X_T$.
So other than eg in the case of binary options / barriers / collars really it is more exotic for $f(X_T)$ not to be continuous - I would say the vast majority of contracts traded are continuous.
Across all of Fixed Income, generally the smallest units of measurement are '0.25 of a basis point', where a 'basis point' is $10^{-4}$. All asset classes have a smallest possible 'tick' size. Strikes on payoffs and in my experience all observations relevant to to evaluating them are necessarily discretised from a measurement perspective, but that's not really the point.
EDIT:
So perhaps more directly to answer your question, you can almost surely assume that the claim is continuous, except when it obviously isn't, as in the short list given above (and apologies for the negligible mathematical content of this post! :))
EDIT 2:
Thanks for the comment Donatien. So, if we raise the bar to the function of our process having to be twice differentiable, I think the below is a relatively simple, (1-dimensional) example that is actually used. Different processes may actually be used in some cases due to preferential properties, (will mention later), but O-U is just fine for our purposes, and in fact the mean-reversion is a useful property, even if perhaps negative sample paths are not in my setting.
So, let $X_T$ be used to define a credit default intensity $\lambda_t$, which parameterises the inhomogenous Poisson default process ($N$=no. of defaults in [$0,t$]) given by: $$\mathbb{P}(N=n)=\frac{\left(\int_0^t \lambda_s ds\right)^n \cdot e^{\int_0^t \lambda_s ds}}{n!}$$
Then the survival probability $S_t:=\mathbb{P}(\tau>t) = \mathbb{P}(N=0)$ ($\tau$ being the default time) is exponentially distributed as per: $$S_t=e^{\int_0^t \lambda_s ds}$$
Now this quantity is twice differentiable in $X_t$ (agreed?). Though this seems an 'embedded' example it's something that is extremely important in applications. Defining a simple payoff at maturity where you receive 1 at maturity contingent on survival and nothing otherwise does the job I think.
Remarks:
The use of O-U is problematic in that it admits negative intensities, as which is the case with processes like Hull-White, however these things still get used. A far better process imo is the CIR++ or JCIR++ processes advanced by Brigo et al (eg in https://arxiv.org/pdf/0911.3331.pdf) which in addition to mean reversion, do not admit negative intensities, and even have analytic formulae for $S_t$ (despite jumps in the JCIR++ case) which is impressive.
Stochastic intensities are important of course where credit volatility and cross factor correlations are important (options / CVAs etc), hence things like the above get used a lot. Hope that's a meaningful (and correct) example even if it's not a documented payoff as such. There might be less embedded examples though.
EDIT 3:
In the spirit of finding an actual payoff satisfying the differentiability condition in addition to the survival probability model given above, this (old) article ('Quadratic Variation - based Payoffs') gives a good summary of at least one that has at least been traded, the variance (volatility) swap (where at term, the realised variances (volatilities) to term are exchanged vs the strike - so like the article says it's really a forward):
http://www.sigmath.es.osaka-u.ac.jp/prob/research/lecture7_2004.pdf
For ease of reading here's an abbreviated version of the payoff:
$$N \cdot \Lambda \cdot \left[ \frac{1}{N}\sum_{i=1}^N \left\{\mathrm{log}\left(\frac{S_i}{S_{i-1}} \right) \right\}^2 \right] - N \cdot K_{var}$$
Where $N$ is notional, $\Lambda$ is an annualisation factor and $K_{var}$ the  strike. I should point out that clearly this payoff is calculated on a lognormal basis however so O-U is at odds with this, however the payoff calculation formalism (necessarily generic) is in a sense independent of the chosen underlying modelling dynamics, although this is not ideal. 
Be interested if any of the above serve your purpose! Cheers.
