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4.3 Derivability and Continuity of A Function At A Point
THEOREM: If a function ‘f ’ is derivable at x, then ‘f ’
is continuous at x.
PROOF: We are given that ‘f’ is derivable at x
\ f ’ (x) = 
REMARK: The converse of this theorem i.e. " If ‘f ’
is continuous at ‘x’ , then ‘f’ is derivable at ‘x’ " is not
true. This can be seen from the following examples.
Example 3
Shows that the function f defined by f(x) = |x| is continuous but
not derivable at x = 0.
We have f (0) = 0 and  |x| = 0. Hence f (x) = f (0) and ‘f ’ is continuous at x = 0.
From Example 1 we have already proved that ‘f ’ is not derivable at x = 0
Example 4
If f (x) = x sin 
, x ¹ 0 and f (0) = 0, show that ‘f ’
is continuous but not derivable at x = 0.
We have f (0) = 0 and
x sin
= (a finite limit) ´
0 = 0 since sin
always lies between -1 and 1 for any value of x.
Therefore -|x| £
x sin£
|x|
But
|x| = 0 and
- |x| = 0 Þ
x sin
= 0
Now f "(0) = 
=  , for the given function.
\ f ’ (0) = 
which does not exist, Hence ‘f ’ is not derivable at x = 0
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