Classification vs. regression for prediction of the sign of a continuous response variable Say I want to predict whether or not a project will be profitable.  In my sample data, the response variable is actually a continuous variable: the $ profit/loss of the project.  
Because my ultimate goal is just a binary classification (profitable project or unprofitable project), should I use a classification technique? Or should I use regression so that I'm not throwing away the additional information a continuous response variable provides?
 A: After reading the comments, I think the following distinction is what's missing from the discussion:
How to model the problem
This has nothing to do with what technique to use. It's the question of what the inputs and outputs are and how to evaluate the result. 
If you truly only care about whether or not our projects are profitable, and the amount by which they are so is absolutely irrelevant, then you should model this as a classification problem. That means you are ultimately optimizing for the expected rate of correct classification (accuracy) or AUC. What this optimization translates to depends on what technique you use. 
All questions of model choice and search algorithms can be approached heuristically (using the arguments that have been put forth in the other answers and comments) but the ultimate proof of the pudding is in the eating. Whatever model you have, you will evaluate by cross validated tests for accuracy, so accuracy is what you optimize.
How to solve the problem
You can use any method you like that fits the classification paradigm. Since you have a continuous y variable, you can do regression on that, and translate to a binary classification. This will likely work well. However, there is no guarantee that the optimal regression model (by sum of squared errors or maximum likelihood or whatever) will also give you the optimal classification model (by accuracy or AUC).
A: I can't think of an example where I would recommend a classification technique when the variable is continuous or ordinal.  After efficiently fitting a continuous model you can use that model to estimate the probability  that $Y$ exceeds any level of interest.  If the model is Gaussian this probability is a function of the predicted mean and the residual standard deviation.
A: Vladimir Vapnik (co-inventor of the Support Vector Machine and leading computational learning theorist) advocates always trying to solve the problem directly, rather than solving some more general problem and then discarding some of the information provided by the solution.  I am generally in agreement with this, so I would suggest a classification approach for the problem as currently posed.  The reason for this is that if we are only interested in classifying a project as profitable or non-profitable, then we are really only interested in the region where profitability is around zero.  If we form a classification model, that is where we will be concentrating our modelling resources.  If we take a regression approach, we may be wasting modelling resources to make small improvements in performance for projects that will either be very profitable or unprofitable, potentially at the expense of improving performance of borderline projects.
Now the reason that I said "as currently posed", is that very few problems actually do involve simple, hard binary classification (optical character recognition would probably be one).  Generally different kinds of misclassification have different costs, or operational class frequencies may be unknown, or variable etc.  In such cases it is better to have a probabilistic classifier, such as logistic regression, rather than an SVM.  If seems to me that for a financial application, we will do better if we know the probability of whether the project will be profitable, and how profitable or otherwise it is likely to be.  We may well be willing to fund a project that has a small chance of being profitable, but massively profitable should it succeed, but not a project that is almost guaranteed to be successful, but which will have such a small profit margin that we would be better off just sticking the money in a savings account.
So Frank and Omri374 are both right! (+1 ;o)
EDIT: To clarify why regression might not always be a good approach for solving a classification problem, here is an example.  Say we have three projects, with profitability $\vec{y} = (-\$1000,+\$1, +\$1000)$, and for each project, we have an explanatory variable that we hope is indicative of profitability, $\vec{x} = (1, 2, 10)$.  If we take a regression approach (with offset), we get regression coefficients $\beta_0 = -800.8288$ and $\beta_1 = 184.8836$ (provided I have done the sums correctly!).  The model then predicts the projects as yielding profits $\hat{y}_1 \approx -\$616$, $\hat{y}_2 \approx -\$431$ and $\hat{y}_3 \approx \$1048$.  Note that the second project is incorrectly predicted as being unprofitable.  If on the other hand, we take a classification approach, and regress instead on $\vec{t} = 2*(y >= 0) - 1$, we get regression coefficients $\beta_0 = -0.2603$ and $\beta_1 = 0.1370$, which scores the three projects as follows: $\hat{t}_1 = -0.1233$, $\hat{t}_2 = 0.0137$ and $\hat{t}_3 = 1.1096$.  So a classification approach correctly classifies project 1 as unprofitable and the other two as being profitable.
The reason why this happens is that a regression approach tries equally hard to minimise the sum of squared errors for each of the data points.  In this case, a lower SSE is obtained by allowing project two to fall on the incorrect side of the decision boundary, in order to achieve lower errors on the other two points.  
So Frank is correct in saying that a regression approach is likely to be a good approach in practice, but if classification actually is the ultimate aim, there are situations where it can perform poorly and a classification approach will perform better.
A: A classification model generally attempts to minimize the sign (error in terms of class) and not the bias.
In case of many outliers, for example, I would prefer using a classification model and not a regression model.
A: I would frame the problem as that of minimizing loss. The question is what is your true loss function? Does an incorrect prediction of profitable when the project lost \$1 cost as much as a prediction of profitable when the project lost \$1000? In that case your loss function is truly binary, and you're better of casting the whole thing as a classification problem. The regression function may still be one of your candidate classifiers, but but you should optimize it with the discrete loss function rather than the continuous one. If you have a more complicated definition of loss, then you should try to formalize it, and see what you get if you take the derivative.
Interestingly, many machine learning methods actually optimize a discrete loss function by approximating with a continuous one, since a discrete loss function provides poor gradients for optimization. So you may end up casting it as a classification problem, since that's your loss function, but then approximating that loss function with the original continuous one.
