unit 1 differential calculus - IGNOU

Mathematics-II
34
f ( x)
(iii) , g ( a)
≠ 0, are continuous at x = a.
g ( x)
(iv) If f (x) is continuous at x = x0 and if g (y) is continuous at
y = y0 = g (x0), then the composite function F (x) = g (f (x)) is
continuous at x = x0.
These results are in fact the immediate corollaries of the corresponding limit
theorems discussed in Section 1.4 also.
If a function is not continuous at a point x0, it is said to be discontinuous at x0.
Example 1.18
sin x
Let f ( x)
= ( x ≠ 0)
. Now f is not defined at x = 0. If we define f (0) = 1
x
sin x
which is same as lim , then f is continuous at x = 0.
x →0
x
y
Example 1.19
The function f defined by
is discontinuous at x = 0.
0
Figure 1.19
⎧ 1 for x > 0
f ( x)
= ⎨
⎩−
1 for x ≤ 0
y
1
0
−1
Figure 1.20
sin x
y =
x
x
x