What class of non linear functions can be modeled by a neural network If machine is viewed as function approximation, what class of functions are modeled by a neural network?
 A: Artificial neural network: computational power (Wikipedia):

The multi-layer perceptron (MLP) is a universal function approximator,
  as proven by the Cybenko theorem. However, the proof is not
  constructive regarding the number of neurons required or the settings
  of the weights.
Work by Hava Siegelmann and Eduardo D. Sontag has provided a proof
  that a specific recurrent architecture with rational valued weights
  (as opposed to full precision real number-valued weights) has the full
  power of a Universal Turing Machine using a finite number of
  neurons and standard linear connections. They have further shown that
  the use of irrational values for weights results in a machine with
  super-Turing power.

Cybenko, G.V. (1989). Approximation by Superpositions of a Sigmoidal function, Mathematics of Control, Signals, and Systems, Vol. 2 pp. 303–314.
Siegelmann, H.T. and Sontag, E.D. (1994). Analog computation via neural networks, Theoretical Computer Science, v. 131, no. 2, pp. 331–360.
A: Actually a three layer neural network can model arbitrary function with the linear and logistic functions, which was proved by Kolmogorov in 1957 (Kolmogorov, Andrei Nikolaevich. "On the representation of continuous functions of many variables by superposition of continuous functions of one variable and addition." Dokl. Akad. Nauk SSSR. Vol. 114. No. 5. 1957.). 
