Now, we'll explore what will happen if we have a pair of continuous functions. So, we have a function which is f of x and another is g of x. Let's suppose that zero belongs to D, and this is the common domain, for them both common domain for f and g, and we know they are continuous at x zero. Can we claim that, if we form the sum or difference or product or even a quotient. Although, in the last case, we need to exclude while given a condition will be given g at x zero, takes a non-zero value. What can we say? Plus minus a product or quotient. So, all in all we have four cases here. Whether they're resulting functions will be continuous at the given point. Yes that's true and that's fairly easy to prove. For example, as an example, we can a choose a product for example, and how to prove that the limit of the product as x tends to x zero equals. In order to prove that, let's recall that, the definition of a limit of a function of many variables is based on the limit of the corresponding sequence. As a sequence of points tends to x zero and we choose any such sequence. We consider whether the product of two sequences has the limit but from the knowledge of a single variable calculus we now given two convergent sequences. The first converges to one limit, the second to another limit, the product or the terms will converge to the product of the limits and that concludes the proof of this statement. As for the quotient it takes and some more efforts should be placed on showing that, if the denominator doesn't equal zero at the x zero point. Then there is a neighborhood or this point where g doesn't equal zero. So, not only the function itself doesn't turn zero at a point but also it doesn't turn in some neighborhood, and that requires some extra statements some work should be done on that. Now, we are approaching a notion of a composite function. Composite function, what it means? I'll start with some examples. For example, a function f, depending on xy, sine x times y. We see that actually there are two functions are involved here, in the definition or such function because we have a function let us be g of xy which is x times y. We have another function which is a function f, is sin t. Now, if we say t equals this function g, which is xy, and we substitute t into here, so we have a couple of functions, this one is an inner function sine, t is an outer function. When we get the function which was defined by the formula, it looks as if that we need to think about the domains of two functions. So, the inner function, now we'll generalize so that was just an example where let's generalize. So, we have a function let it be t equals g of xy, now f of xy will be a function h of t. Okay? So, if we substitute this inner function into h of t, we get this composite function f. In order to do that, we need to be sure that the image that the range over g range of g. Remember, we get the range of g when we choose any point from the domain of the inner function, we substitute the values of the point the point into the argument. As a result we get t value and all such t-values, they form the range of g. So, we need to be sure that the domain or the outer function which is a function of one variable includes the range of g. So, here I can draw a diagram, so we have D. D is domain of g. Now, here we have range of g, these are the t-values, and all these values belong to the domain of h of t. Then we're sure that we can choose any point from D, substitute into the argument of g and calculate, firstly t-value then substitute t-values into h and get the resulting value of the function f. If not, we maybe in a complicated situation look at this, let's consider a function which is provided by the formula minus one, minus x squared minus y squared. What about the domain of the inner function? We can substitute for x and y any values. So, the domain D coincides with whole a coordinate plane r two, but the range of this inner function, so this is our g of xy, this one, g xy. From it's the part of the real number line, but g takes the values which are less than or equal to negative one. Since we are unable to find the square root of negative values, social functions simply doesn't exist. So, the domain for this function is the empty set. Now, we will be interested in effect, given a composite function based on a couple of functions in a function is a function of many variables like g and the outer function is a function of one variable only h of t. Can we claim that the composite function based on these two functions, will be a continuous function given the knowledge that both g of xy and h of t are continuous at the corresponding points? We'll see.
