E. Relation between Continuity & Differentiability
If a function f is derivable at x then f is continuous at x.
If f(x) is derivable for every point of its domain, then it is continuous in that domain .
The converse of the above result is not true :
"If f is continuous at x, then f may or maynot be derivable at x"
The functions f(x) =^{ } & g(0) = 0 are continuous at x = 0 but not derivable at x = 0.
Remark :
(a) Let f'_{+}(a) = p & f'_(a) = q where p & q are finite then :
(i) p = q ⇒ f is derivable at x = a ⇒ f is continuous at x = a.
(ii) p ≠ q ⇒ f is not derivable at x = a but f is continuous at x = a
Differentiable ⇒ Continuous ; Nondifferentiable Discontinuous
But Discontinuous ⇒ Nondifferentiable .
(b) If a function f is not differentiable but is continuous at x = a it geometrically implies a sharp corner at x = a.
Ex.15 If f(x) = , then find the value of k so that f(x) becomes continuous at x = 0. Hence, find all the points where the functions is nondifferentiable.
Sol. From the graph of f(x) it is clear that for the function to be continuous only possible value of k is 1.
Points of nondifferentiability are x = 0, ±1.
Ex.16 If f(x) = where [.] denotes the greatest integer function.
Discuss the continuity and differentiability of f(x) in [0, 2).
Sol. Since 1 x  1 < 1 then [x^{2}  2x] = [(x  1)^{2}  1] = [(x  1)^{2}]  1 = 0  1 = 1
Graph of f(x) :
It is clear from the graph that f(x) is discontinuous at x = 1 and not differentiable at x =1/2,and x= 1
Further details are as follows : ⇒
Hence, which shows f(x) is not differentiable at x = 1/2 (as RHD = 4 and LHD = –4) and x = 1 (as RHD = 0 and LHD = 8). Therefore, f(x) is differentiable, for x ∈ [0, 2)  {1/2, 1}
Ex.17 Suppose f (x) = . If f '' (1) exist then find the value of a^{2} + b^{2} + c^{2}.
Sol. For continuity at x = 1 we leave f (1^{–}) = 1 and f (1^{+}) = a + b + c
a + b + c = 1 ....(1)
for continuity of f ' (x) at x = 1 f ' (1^{–}) = 3; f ' (1^{+}) = 2a + b
hence 2a + b = 3 ....(2)
f '' (1–) = 6; f '' (1+) = 2a for continuity of f '' (x) 2a = 6 ⇒ a = 3
from (2), b = – 3 ; c = 1. Hence a = 3, b = – 3 ; c = 1
Ex.18 Check the differentiability of the function f(x) = max {sin^{1} sin x, cos^{1} sin x}.
Sol. sin^{1} sin x is periodic with period ⇒ sin^{1} sin x =
Also cos^{1} sin x =  sin^{1} sin x
⇒ f(x) is not differentiable at
⇒ f(x) is not differentiable at
Ex.19 Find the interval of values of k for which the function f(x) = x^{2} + (k  1) x  k is non differentiable at five points.
Sol.
f(x) = x^{2} + (k – 1) x – k = (x – 1) (x + k)
Also f(x) is an even function and f(x) is not differentiable at five points.
So (x – 1) (x + k) is non differentiable for two positive values of x.
⇒ Both the roots of (x – 1) (x + k) = 0 are positive.
⇒ k < 0 ⇒ k ∈ (–∝, 0).
Definition : A function f is differentiable at a if f'(a) exists. It is differentiable on an open interval (a,b) [or (a, ∝) or (–∝, a) or (– ∝, ∝)] if it is differentiable at every number in the interval.
Derivability Over An Interval : f(x) is said to be derivable over an interval if it is derivable at each & every point of the interval. f(x) is said to be derivable over the closed interval [a, b] if :
(i) for the points a and b, f'(a^{+}) & f'(b ^{}) exist &
(ii) for any point c such that a < c < b, f'(c^{+}) & f'(c ^{}) exist & are equal .
204 videos288 docs139 tests

1. What is the definition of continuity in mathematics? 
2. Can a function be continuous but not differentiable? 
3. Is every differentiable function continuous? 
4. How can we determine if a function is differentiable at a specific point? 
5. Can a function be differentiable but not continuous? 

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