TABLE OF CONTENTS Limits of functions are defined in terms of limits of sequences. With a function f continuous on an interval we associate the intuitive idea of the graph f being drawn without lifting the pencil from the drawing paper. Mathematical treatment of continuity starts with the definition of a function continuous at a point; this definition is given here in terms of a limit of a function at a point. In this chapter we shall develop the theory of limits of functions, study continuous functions, and particularly functions continuous on closed bounded intervals. At the end of the chapter we touch upon the concept of limit superior and inferior of a function.
Looking at the graph of 
(Figure 12.1), it is natural to say that the function value approaches 1 as x approaches 0 from the right. Formally we define:
A function f, with dom f ? 
, is said to have a limit l at 
from the right if for every sequence n ? x n for which x n ? and x n> it follows that f( x n) ? l. If f has a limit l at from the right, we write 
.
If the condition x n> is replaced by x n< one obtains the definition of the limit of f at from...
