Page 156 - DMTH401_REAL ANALYSIS

Real Analysis

Notes Leaving the point ‘a’, we can write it as
0 < (x – a) <   |f(x) – f(a)| < E.
This implies the existence, of lim f(x) and that lim f(x) = f(a).
x® a x® a
Note that  in the definition 2, in general, depends on the given function f,  and the point a. Also
|x – a| <  if and only if a –  < x < a +  and ]a – , a + [ is an open interval containing a. Similarly
|f(x) – f(a)| <  if and only if

f(a) –  < f(x) < f(a) + E.
We see that f is continuous at a point a, if corresponding to a given (open) -neighbourhood U of
f(a) there exists a (open) -neighbourhood V of a such that f(V)  U. Observe that this is the same
as x  V  f(x)  U. This formulation of the continuity at a is more useful to generalise this
definition to more general situations in Higher Mathematics.

A function f is said to be continuous on a set S if it is continuous at every point of the set S. It is
clear that a constant function defined on S is continuous on S.

Example: Examine the continuity of the following functions:
(i) The absolute value (Modulus) function,
(ii) The signum function.
Solution:
(i) You know that the absolute value function

f: R ® R is defined as f(x) = |x|, V x  R.
The function is continuous at every point x  R. For given  > 0, we can choose  =  itself.
If a  R be any point them |x – a| <  =  implies that
|f(x) – f(a)| = ||x| – |a||  |x – a| < .

(ii) The signum function, as you know a function f: R ® R defined as
f(x) = 1 if x > 0
= 0 if x = 0
= –1 if x < 0
This function is not continuous at the point x = 0. We have already seen that f(0+) = 1, f(0–) = –1.
Since f(0+)  f(0–), lim f(x) does not exist and consequently the function is not continuous at
x – 0
x = 0. For every point x  0 the function f is continuous. This is easily seen from the graph of
the function f. There is a jump at the point x = 0 in the values of f(x) defined in a neighbourhood
of 0.
Note that if f: R ® R is defined as,
f(x) = 1 if x  0.

= –1 if x < 0.
then, it is easy to see that this function is continuous from the right at x = 0 but not from the left.
It is continuous at every point x  0.
Similarly, if f is defined by f(x) = 1 if x > 0
= –1 if x  0
then f is continuous from the left at x = 0 but not from the right.

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