What is the input space of a neural network (or other supervised learning algorithms)? While training the neural network (or any other supervised learning algorithms), we supply input variables and corresponding outputs. The input variables can be continuous or discrete (binary in many cases). 

What happens if after training with a binary input data, we supply a continuous value for the same input at the time of evaluation? Does the algorithm internally treat all variables as continuous variables? 

For example, suppose one of the inputs is Young/Old encoded in the form of 0/1 in the training dataset. What happens if we supply a value of, say, 0.2 at the prediction stage? Does it/should it make any sense to the network?
I guess it depends on the algorithm, but linear models, as well as neural networks, treat all variables as continuous. The algorithm will not explode or anything if you supply 0.2 at prediction stage. However, your algorithm is trained on data. The algorithm can at best do what it has learnt from the training data. For this reason, do not expect anything meaningful when you feed an example with a value that has not been seen in the whole training set, or that it does not follow the training set distribution.