Page 243 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes Thus, |x – 2| is continuous at x = 2
After considering the above results, we state below some properties of continuous functions.
If f (x) and g (x) are two functions which are continuous at a point x = a, then,
(a) C f(x) is continuous at x = a, where C is a constant.
(b) f(x) g(x) is continuous at x = a.
(c) f(x) . g(x) is continuous at x = a.

(d) f (x)/g (x) is continuous at x = a, provided g (a) 0.
(e) |f(x)| is continuous at x = a.
Thus every constant function is a continues function

1. Prove that tan x is continuous when

2. Let f (x) = Show that f is continuous at 1.

8.4.2 Important Result of Constant Function

By using the properties mentioned above, we shall now discuss some important results on
continuity.
1. Consider the function f(x) = px + q, x R
The domain of this functions is the set of real numbers. Let a be any arbitary real number.
Taking limit of both sides of (i), we have

px + q is continuous at x = a.
2
Similarly, if we consider f(x) = 5x + 2x + 3, we can show that it is a continuous function.
In general

where a + a + a … a are constants and n is a non-negative integer,
0 1 2 n
we can show that are all continuos at a point x = c (where c is any
real number) and by property 2, their sum is also continuous at x = c.
f (x) is continuous at any point c.
Hence every polynomial function is continuous at every point.

2. Consider a function f(x) = , f(x) is not defined when x – 5 = 0 i.e, at x = 5.

Since (x + 1) and (x + 3) are both continuous, we can say that (x + 1) (x + 3) is also continuous.
[Using property 3]
Denominator of the function f (x), i.e., (x – 5) is also continuous.

Using the property 4, we can say that the function is continuous at all
points except at x = 5.

236 LOVELY PROFESSIONAL UNIVERSITY