About the Question 3 in this midterm , there are 3 parts need solving. Although the forms are different, fundamentally speaking ,they are all related to the definition of  continuity. So the key to solve this kind of problem is to understand what is the continuity.

continuity means “constancy “, let us think about what if a function is continuous in its domain , it should be complete, without break ;  it should be smooth, without jump.

Look back to Question 3, about (a) we use the definition of continuity to define g(x) to be continuous at a , and about (b) , we need  to define what it means for a function f(w) to be continuous at g(a), we can just represent g(a) as a certain number just like a, and now we can find it has the same form as the question (a) . About (c) , although the form looks more complicated than (a)(b), the key to solve the problem is also simplification, and we also use the knowledge about the definition of limit to combine g(x) with f(w) to get the answer.

About the Question 3 in this midterm , there are 3 parts need solving. Although the forms are different, fundamentally speaking ,they are all related to the definition of  continuity. So the key to solve this kind of problem is to understand what is the continuity.

continuity means “constancy “, let us think about what if a function is continuous in its domain , it should be complete, without break ;  it should be smooth, without jump.

And now give the real definition of continuity

A function f is continuous at x=a provided all three of the following are true:

(i) the function f is defined at a

(ii) the limit of f as x approaches a from the right-hand and left-hand limits exist and are equal, and

(iii) the limit of f as x approaches a is equal to f(a).

The function must satisfy all three requirements, if one of these is not fit, the function is not continuous.

Look back to Question 3, about (a) we use the definition of continuity to define g(x) to be continuous at a , and about (b) , we need  to define what it means for a function f(w) to be continuous at g(a), we can just represent g(a) as a certain number just like a, and now we can find it has the same form as the question (a) . About (c) , although the form looks more complicated than (a)(b), the key to solve the problem is also simplification, and we also use the knowledge about the definition of limit to combine g(x) with f(w) to get the answer.