The law of supply shows a positive relationship between price and quantity However, a firm producing at break even level of output would have to increase the quantity supplied if the price of the commodity goes down in order to break even. Does that mean that a firm producing at break even level is an exception to the law or that the analysis contradicts the law of supply ? ( please correct me if there are any inconsistencies in the logic)
The short answer is no.
The law of supply only states that increases in price result in higher quantities supplied, ceteris paribus. Break even is the point where Revenue = Costs.
Your confusion might come from this sentence:
a firm producing at break even level of output would have to increase the quantity supplied if the price of the commodity goes down in order to break even.
If the price lowers, the firm will reduce its quantity supplied. That's it. The specific quantity depends on the variable and fixed costs. It's not mentioned that the company has to break even - think of a monopolist for example.
The logic of the OP implies that all costs are fixed. In the very short-term one could say that this holds for many firms: a firm cannot leave rented buildings immediately, it can not fire employees immediately etc. Namely "costs" have the form of financial commitments.
But in the mid term (and I am talking a few weeks to a few months, no more), some of the costs are in reality variable. Then if at the original break-even point average cost was increasing, the reduction of price will lead the firm to a lower quantity (where it will have lower average cost), that may even bring in profits.
And the "Law of Supply" refers not to the very short term, but rather to the mid- and long-term behavior.