Page 240 - DMTH201_Basic Mathematics-1

Unit 8: Continuity

Notes
b – a = …(i)
n n

and that r is defined by
n
r = …(ii)
n
In order for the error to be smaller than ,

–n
= 2 (b – a) < …(iii)
Taking the natural logarithm of both sides then gives
–n ln2 < ln – ln (b – a), …(iv)

so from 1,2,3 and 4 the result is

8.4 Function at a Point

So far, we have considered only those functions which are continuous. Now we shall discuss
some examples of functions which may or may not be continuous.

x
Example: Show that the function f(x) = e is a continuous function.
Solution:
x
Domain of e is R. Let a R. where ‘a’ is arbitrary.
=

=

=

=

=
…(i)
= …(ii)

f(a) =
From (i) and (ii),

f(x) is continuous at x = a
x
Since a is arbitary, e is a continuous function.
8.4.1 Properties of Continuos Function

1. Consider the function f (x) = 4. Graph of the function f (x) = 4 is shown in the Figure 8.1.
From the graph, we see that the function is continuous. In general, all constant functions are
continuous.

LOVELY PROFESSIONAL UNIVERSITY 233