Offset needed in regression when response is continuous? I know that for poisson regressions on count data that originate from different sampling "sizes", i.e. different volumes, areas etc, require an offset in order to adjust for the different sizes.  
However, in Zuur et al. (2009) Mixed Effects Models in R in read on page 198 (ch.8.3.1.)

One option is to use the density Ni/Vi as the response variable and
  work with a Gaussian distribution, but if the volumes differ
  considerably per site, then this is a poor approach as it ignores the
  differences in volumes.

In my case, I have counts of harbor porpoise sightings on different sized areas. But, the counts were transformed to densities. Yet, the areas on which the densities were calcualted from are very different.   
My Question now:  Can I use an offset for a continous response (actually I use the tweedie distribution since I have more than 60% zeros in the data).  
One additonal thing: I compared models with offset and without using AIC. The one with the offset(log(area+1))  was "best".
 A: Short answer: yes, you can use an offset in any GLM. Longer answer: you should be clear on what it is actually doing.
An offset simply adds some fixed value to the linear predictor:
$g(E[Y]) = \eta = \sum_{i=1}^p X_i \beta_i + \text{offset}$
where $g$ is the link function. In other words, you can think of it as a term in the model whose coefficient is fixed, rather than estimated.
The most common application (and the one you seem to be thinking of) occurs when the response is thought to be directly proportional to some exposure variable, and a log link function is used. In this case, the log-exposure is used as the offset:
$\log(E[Y]) = \sum_{i=1}^p X_i \beta_i + \log(\text{exposure})$
which gives:
$E[Y] = \text{exposure} \times \exp\left\{\sum_{i=1}^p X_i \beta_i\right\}$
Note the assumption: if exposure is twice as large, so will the expected response. Other than that, any response distribution should work (though only for the log-link).
However I would be skeptical about using area + 1 as the exposure variable: was there any reason you chose this rather than just area?
