The cross-entropy error function in neural networks

In the MNIST For ML Beginners they define cross-entropy as $$H_{y'} (y) := - \sum_{i} y_{i}' \log (y_i)$$ $y_i$ is the predicted probability value for class $i$ and $y_i'$ is the true probability for that class. Question 1 Isn't it a problem that $y_i$ (in $\log(y_i)$) could be 0? This would mean that we have a really bad classifier, of course. But think of an error in our dataset, e.g. an "obviou