Supply Curve of Firm and Industry:

For a given price, the firm will produce only that much of output where MR = MC. Therefore, MC curve lying above the average variable cost (AVC) would give the supply curve of the firm. The portion of MC lying below the AVC would not represent the supply curve because in that region the firm would shut down. The reason for it is shown with the help of the following proof :
Suppose p< AVC

⇒ pq < (AVC)q

⇒ TR < TVC

⇒ TR < TC - TFC

⇒ TC - TR > TFC

⇒ the loss incurred by the firm is greater than the total fixed cost, so that if the firm shuts down, it will have to incur only the fixed cost.

Supply Curve of the Firm 

Diagrammatically, the supply curve of the firm would be given by the rising part of the MC curve lying above the AVC curve. Below the AVC curve the supply curve would coincide with the vertical axis. Therefore, the supply curve would be broken one as shown in Figure. 

Mathematical Representation:
      Let the cost function of the i^th
 firm be C = C_i (q_i)  
      The marginal cost, MC = ∂C_i (q_i)/ ∂q_i  = C′_i (q_i)
       Given the market price (p) the supply function of the firm is given by,
                    S_i = S_i (P) for p ≥ min AVC
                        = 0       for p < min AVC  
The aggregate supply function or the supply function of the industry is given by  
                 S = ∑ S_i (p) = S (p)