Why deep feedforward neural networks are restricted to solve classification problems?
There is regression by neural networks. For example, have a look at this analysis (Deep Regression), RBF neural networks, and General regression neural networks. Also, linear regression can be implemented using a neural network.
If using neural networks for interpolation, this may work very good. However, if regression is used to extrapolate, results may be unsatisfactory.
Note also that the universal approximation theorems say only that a contineous function can be approximated within a compact set (thus in particular bounded set!).