unit 1 differential calculus - IGNOU

Remark
If the interval is closed, the limit in the continuity test over the interval is
two-sided at an interior point and the appropriate one-sided at the end
points.
Example 1.16
Prove that f (x) = sin x is continuous at x = 0.
Solution
(i) f (0) = sin 0 = 0
(ii) f ( x)
= sin x = 0 and
lim
x →0
lim
x →0
(iii) f ( x)
= f (0) = 0
lim
x →0
Therefore, f (x) = sin x is continuous at x = 0. (In fact sin x and cos x are
continuous at each real x.)
Example 1.17
Examine the continuity of the function .
1
f ( x)
=
x
Solution
The function is defined for all non-zero real value of x. It is not defined at
x = 0.
Figure 1.18
1
Also lim f ( x)
= = f ( x0),
if x0
≠ 0
x → x x
0
0
Therefore, the function is continuous for all real x ≠ 0. It fails to be
continuous at x = 0 (why)? (Figure 1.18). This function is continuous on
any interval which does not include x = 0 as an element of it.
Theorem 4
If the functions f (x) and g (x) are continuous at x = a, then
(i) f (x) ± g (x),
(ii) f (x) g (x), and
y
0
1
y =
x
x
Differential Calculus
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