 Hi and welcome to the session. Today we will learn about continuity. First of all let us understand graphically with how to find out whether a function is continuous or not. So if we can draw the graph of the function without lifting the pen from the plane of the paper then the function is said to be continuous. For example here we can draw this graph without lifting our pen from the plane of the paper so that means the function represented by this graph is continuous. And in this we cannot draw the graph without lifting our pen from the plane of the paper that means the function represented by this graph is not continuous. So now let us understand this mathematically. Suppose f is a real function on a subset of the real numbers and let we point in the domain of f then is continuous at c if limit of f of x where x tends to c is equal to f of c. We can also say that if at x equals to c left hand limit right hand limit and value of the function that is f of c exists and are all equal to each other then the function f is continuous. So in other words we can say that a function f is continuous at x equals to c if the function is defined at x equals to c and if the limit of f of x where x tends to c is equal to f of c that is the value of the function at x equals to c. Now if f is not continuous at point c then we say that the function f is discontinuous at point c. The point c is called the point of discontinuity the function f. Now let us take one example here we are given the function f as f of x equal to 4x cube plus 3x square minus 5 and we need to check the continuity of the function f at point x equal to 1. Now here clearly the function f is defined at point x equal to 1 and the value of the function f at point x equal to 1 that is f of 1 is equal to 4 into 1 cube plus 3 into 1 square minus 5 which is equal to 2 the limit of f of x where x tends to 1 is equal to limit x tends to 1 of 4x cube plus 3x square minus 5 which is equal to 2 that means limit of f of x where x tends to 1 is equal to f of 1 that is the value of the function f at point x equal to 1. So this implies that the function f is continuous at point x equal to 1. Now a real function is said to be continuous if it is continuous at every point in the domain that is suppose is a function defined on a closed interval a comma b then for f to be continuous it needs to be continuous at every point in a closed interval a comma b including the end points a at b continuity of f at a means limit f of x where x tends to a plus is equal to f of a continuity of f at b limit of f of x where x tends to b minus is equal to f of b. Now there is one important point for you to remember that every polynomial function is continuous. Let's move on to our next topic that is algebra continuous functions in this first important result states and g are two real functions continuous at a real number c then g is continuous at x equal to c f minus g is continuous at x equal to c into g is continuous at x equal to c and lastly upon g is also continuous at x equal to c. Now there is a condition attached to it that is provided g of c is not equal to 0. Now there is a second result and g are real valued functions such that the composite function f of g is defined at c if g is continuous at c and if f is continuous g of c then the composite function f of g is continuous now we will take one example for this but before that you need to remember few important points that is every constant function is continuous identity function is continuous every rational function is continuous cosine functions continuous now here we have an example we need to prove that the function given by f of x is continuous f of x is equal to cos of x square so let us suppose that g of x is defined as x is square and the function h of x is defined as cos x now x square is a polynomial and we know that every polynomial function is continuous so from this we can say that the function g is continuous also we know that cosine function is continuous that means the function h is also continuous so that means the function h of g will be continuous of g is equal to h of g of x which is equal to h of now g of x is x is square so here we will get x is square so this is h of x is square now h of x is cos x so h of x is square will be cos x is square which is equal to f of x now the composite function h of g is continuous which is equal to f of x that means the function f is also continuous so with this we finish this session hope you must have understood all the concepts goodbye take care and have a nice day