 Hi and welcome to the session. I'm Shashi and I'm going to help you with the following question. Question says, examine the continuity of function f where function f is defined by fx is equal to sine x minus cos x, if x is not equal to 0, fx is equal to minus 1, if x is equal to 0. 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 start with the solution now. We are given function f is given by fx is equal to sine x minus cos x, if x is not equal to 0 and fx is equal to minus 1, if x is equal to 0. Clearly we can see function is defined at every real number. Here it is defined for all the real values of x 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 of x less than and greater than 0. We know fx is equal to sine x minus cos x, if x is not equal to 0. Now we know sine function is continuous at every real number and cosine function is also continuous at every real number. Difference of two continuous functions is also continuous. So we can say function f is continuous at every real number greater than and less than 0. Let us discuss continuity of the function at x is equal to 0. We know function is defined at x is equal to 0 since it is defined at every real number. Now 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 sine x minus cos x. We know for x slightly less than and greater than 0 fx is equal to sine x minus cos x. Now this limit is equal to sine 0 minus cos 0. We know sine 0 is equal to 0 and cos 0 is equal to 1. So we get 0 minus 1 and our required answer is minus 1. So we get limit of x tending to 0 fx is equal to minus 1. Now let us find out value of the function at x is equal to 0. We know f0 is equal to minus 1. Now clearly we can see limit of the function is equal to value of the function at x is equal to 0. So we can write limit of x tending to 0 fx is equal to f0 is equal to minus 1. Now this implies given function f is continuous at x is equal to 0. We have already shown that given function f is continuous at every real number greater than and less than 0. And here we have shown that given function f is continuous at x is equal to 0. Now from these two statements we get function f is continuous at every real number. So our required answer is 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.