 Hi and welcome to the session. Let us discuss the following question. Question says, determine if function f defined by fx is equal to x square sine 1 upon x, if x is not equal to 0, fx is equal to 0, if x is equal to 0 is continuous function. First of all, let us understand that function f is continuous at x is equal to a, if function is defined at x is equal to a, or we can say f a exists and limit of the function is equal to value of the function at x is equal to a. This is the key idea to solve the given question. Let us now start with the solution. We are given function f is given by fx is equal to x square multiplied by sine 1 upon x, if x is not equal to 0, and fx is equal to 0, if x is equal to 0. Now clearly we can see given function f is defined at every real number. Here it is defined at every real number greater than and less than 0, and here it is defined for x is equal to 0. So we can say function f is defined at every real number. Now let us discuss continuity of the function for all the values greater than and less than 0. Now we are given fx is equal to x square multiplied by sine 1 upon x, if x is not equal to 0. Now clearly we can see this is a polynomial function and polynomial function is continuous at every real number. This is a sine function and sine function is also continuous at every real number. Now product of the two continuous functions is also continuous. So we can say function f is continuous at every real number less than and greater than 0. Now let us discuss the continuity of the function at x is equal to 0. Now we know function f is defined at x is equal to 0. Now first of all let us find out limit of the function at x is equal to 0. So we can write limit of x tending to 0 fx is equal to limit of x tending to 0 x square multiplied by sine 1 upon x. We know for x slightly less and greater than 0 fx is equal to x square multiplied by sine 1 upon x. Now this limit is equal to 0 multiplied by some value between minus 1 and 1. We know for limit of x tending to 0 sine 1 upon x oscillates between minus 1 and 1. Now we know 0 multiplied by any number is equal to 0 so we get 0. Now let us find out value of the function at x is equal to 0 that is f0 f0 is equal to 0. Now clearly we can see limit of the function is equal to value of the function at x is equal to 0. Now this implies given function f is continuous at x is equal to 0. We have already shown that function f is continuous at every real number greater than and less than 0. And here we have shown that function f is continuous at x is equal to 0. Now these two statements imply that function f is continuous at every real number. So our required answer is yes function f is continuous for all x belonging to real numbers. This completes the session. Hope you understood the session. Take care and have a nice day.