Why do activation functions have to be monotonic?
The monotonicity criterion helps the neural network to converge easier into an more accurate classifier. 
See this stackexchange answer and wikipedia article for further details and reasons.

However, the monotonicity criterion is not mandatory for an activation function - It is also possible to train neural nets with non-monotonic activation functions. It just gets harder to optimize the neural network. 
See Yoshua Bengio's answer.