Why must a nonlinear activation function be used in a backpropagation neural network?

The purpose of the activation function is to introduce non-linearity into the network

in turn, this allows you to model a response variable (aka target variable, class label, or score) that varies non-linearly with its explanatory variables

non-linear means that the output cannot be reproduced from a linear combination of the inputs (which is not the same as output that renders to a straight line--the word for this is affine).

another way to think of it: without a non-linear activation function in the network, a NN, no matter how many layers it had, would behave just like a single-layer perceptron, because summing these layers would give you just another linear function (see definition just above).

>>> in_vec = NP.random.rand(10)
>>> in_vec
  array([ 0.94,  0.61,  0.65,  0.  ,  0.77,  0.99,  0.35,  0.81,  0.46,  0.59])

>>> # common activation function, hyperbolic tangent
>>> out_vec = NP.tanh(in_vec)
>>> out_vec
 array([ 0.74,  0.54,  0.57,  0.  ,  0.65,  0.76,  0.34,  0.67,  0.43,  0.53])

A common activation function used in backprop (hyperbolic tangent) evaluated from -2 to 2:

A linear activation function can be used, however on very limited occasions. In fact to understand activation functions better it is important to look at the ordinary least-square or simply the linear regression. A linear regression aims at finding the optimal weights that result in minimal vertical effect between the explanatory and target variables, when combined with the input. In short, if the expected output reflects the linear regression as shown below then linear activation functions can be used: (Top Figure). But as in the second figure below linear function will not produce the desired results:(Middle figure). However, a non-linear function as shown below would produce the desired results:

Activation functions cannot be linear because neural networks with a linear activation function are effective only one layer deep, regardless of how complex their architecture is. Input to networks is usually linear transformation (input * weight), but real world and problems are non-linear. To make the incoming data nonlinear, we use nonlinear mapping called activation function. An activation function is a decision making function that determines the presence of a particular neural feature. It is mapped between 0 and 1, where zero means absence of the feature, while one means its presence. Unfortunately, the small changes occurring in the weights cannot be reflected in the activation values because it can only take either 0 or 1. Therefore, nonlinear functions must be continuous and differentiable between this range. A neural network must be able to take any input from -infinity to +infinite, but it should be able to map it to an output that ranges between {0,1} or between {-1,1} in some cases - thus the need for activation function. Non-linearity is needed in activation functions because its aim in a neural network is to produce a nonlinear decision boundary via non-linear combinations of the weight and inputs.

If we only allow linear activation functions in a neural network, the output will just be a linear transformation of the input, which is not enough to form a universal function approximator. Such a network can just be represented as a matrix multiplication, and you would not be able to obtain very interesting behaviors from such a network.

The same thing goes for the case where all neurons have affine activation functions (i.e. an activation function on the form f(x) = a*x + c, where a and c are constants, which is a generalization of linear activation functions), which will just result in an affine transformation from input to output, which is not very exciting either.

A neural network may very well contain neurons with linear activation functions, such as in the output layer, but these require the company of neurons with a non-linear activation function in other parts of the network.

Note: An interesting exception is DeepMind's synthetic gradients, for which they use a small neural network to predict the gradient in the backpropagation pass given the activation values, and they find that they can get away with using a neural network with no hidden layers and with only linear activations.