Page 158 - DMTH401_REAL ANALYSIS

Real Analysis

Notes Conversely, let us suppose that whenever x converges to a, f(x ) converges to f(a). Then we have
n n
to prove that f is continuous at a. For this, we have to show that corresponding to an E > 0, there
exists some  > 0 such that
|f(x) – f(a)|  , whenever |x – a| < .
If not, i.e., if f is not continuous at a, then there exists an E > 0 such that whatever  > 0 we take
there exists an x such that

|x – a| <  but |f(x ) – f(a)|  .
 
By taking  = 1, 1/2, 1/3,.... in succession we get a sequence (x ), where x = x for  = 1/n, such that
n n 
|f(x ) – f(a)|  . The sequence (x ) converges to a. For, if m > 0, these exists M such that 1/n < m
n n
for n  M and therefore |x – a| < m for n  M. But f(x ) does not converge to f(a), a contradiction
n n
to our hypothesis. This completes the proof of the theorem.
Theorem 2 is sometimes used as a definition of the continuity of a function in terms of the
convergent sequences. This is popularly known as the Sequential Definition of Continuity
which we state as follows:
Definition 3: Sequential Continuity of a Function
Let f be a real-valued function whose domain is a subset of the set R. The function f is said to be
continuous at a point a if, for every sequence (x ) in the domain of f converging to a, we have,
n
lim f(X ) = f(a)
n ® ¥ n
The next example illustrates this definition.

Example: Let f: R ® R be defined as

2
f(x) = 2x + 1, " x  R
Prove that f is continuous on R by using the sequential definition of the continuity of a function.

Solution: Suppose (x ) is a sequence which converges to a point ‘a’ of R. Then, we have
n
lim f(x ) = lim (2x + 1) = 2( lim x ) + 1 = 2a + 1 = f(a)
2
2
2
n ® ¥ n n ® ¥ n n – m n
This shows that f is continuous at a point a  R. Since a is an arbitrary element of R, therefore, f
is continuous everywhere on R.

Task Prove by sequential definition of continuity that the function f : R ® R defined by
f(x) = x is continuous at x = 0.

11.2 Algebra of Continuous Functions

As we have proved limit theorems for sum, difference, product, etc. of two functions, we have
similar results for continuous functions also. These algebra operations on the class of continuous
functions can be deduced from the corresponding theorems on limits of functions, using the
limit definition of continuity. We leave this deduction as an exercise for you. However, we give
a formal proof of these algebraic operations by another method which illustrates the use of
Theorem 2. We prove the following theorem:

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