Usually in economic theory we deal with explicitly defined functions, or simply implicit functions. Like in microeconomics, the output of a firm is given by a function which depends on the factors of production, labor and capital. Or the cost function for example, also it depends on the value of the output, but quite often, we have to find a function, for instance, let's consider a perfectly competitive firm that maximizes its profit and the profit Pi is given by the formula, the price or the output per unit times the output minus the cost function CY. So, this is a problem of maximizing the output for the values of the output greater or equal to zero. It's clear how to solve this problem, by differentiation. We have equation, we call it the first-order condition. This is an ordinary derivative and we view this equation as equation defining the output of the firm and we call if we are able to solve it for Y, then Y is a function of the price alone. So, here the price is exogenously given and the firm is a price taker. So, the behavior of the firm doesn't affect the price. So, the question is, whether the output exists as a function of the price. So, in order to formulate the problem in general, let's consider a purely mathematical example. This is an equation of a unit circle and here I draw the graph. This is our unit circle. These are the axes. Here we have 0.1, negative one. The question is, whether it's possible to solve for Y. So, we're looking for a function, Y is a function of X. When I ask the question how to find Y of X, usually the answer is, it's very simple, because we can solve for Y by firstly moving X squared to the right hand side and after that choosing two signs either plus or minus in front of the square root. But that will be a wrong answer. Why? Because we're looking for a function and according to the definition of functions, a function takes one unique value of Y for any X from the domain. Here, we have a combination of signs and we have a difficulty here, what sign to choose. But, if I restate the question, asking whether it's possible to solve this equation in terms of Y of X. Still we're looking for this function, near that point. So, I'm choosing a point which belongs to this unit circle, these are the coordinates. To be more concrete let me choose the coordinates like one over the square root of two that's 4Y0 and one over the square root of two that's 4Y0. And I repeat my question, whether it's possible to find a function YX which satisfies firstly, the given equation and secondly, if we substitute instead of X one over the square root of two, the value of Y will be the same. Then immediately, the answer will follow, we need to choose the plus sign then. That's because the given point belongs to the upper semicircle. If I change the point and choose another which belongs to the lower semicircle, then the choice of the sign will be minus. But this example tells us that in order to find the function as a solution of some equation, we need to be more specific in terms of a point which belongs to the equation. So, we would like to solve this equation in order to find Y as a function of X. Such a function which is defined by an equation is called an implicit function, defined by a particular equation or simply implicit function. By its nature, the solution will have a local specifics that means that we need to indicate a point within the neighborhood of which we are looking for this implicit function. It would be very difficult or simply impossible to answer the question for a role interval of X values stretching from negative one to positive one. So, this result, how to find an implicit function defined by an equation is a local in its nature. So, it would be impossible to get a solution for this implicit function for all values of X from some interval for example. But in a neighborhood of a given point it's quite possible to do so. It's time to state a theorem. We should provide conditions under which the implicit function will exist.