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Relation Between Continuity & Differentiability | Mathematics (Maths) Class 12 - JEE PDF Download

E. Relation between Continuity & Differentiability

If a function f is derivable at x then f is continuous at x.

Relation Between Continuity & Differentiability | Mathematics (Maths) Class 12 - JEE

Relation Between Continuity & Differentiability | Mathematics (Maths) Class 12 - JEE

Relation Between Continuity & Differentiability | Mathematics (Maths) Class 12 - JEE

Relation Between Continuity & Differentiability | Mathematics (Maths) Class 12 - JEE

Relation Between Continuity & Differentiability | Mathematics (Maths) Class 12 - JEE

Relation Between Continuity & Differentiability | Mathematics (Maths) Class 12 - JEE

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) = Relation Between Continuity & Differentiability | Mathematics (Maths) Class 12 - JEE & 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 ; Non-differentiable Relation Between Continuity & Differentiability | Mathematics (Maths) Class 12 - JEE Discontinuous

But Discontinuous ⇒ Non-differentiable .

(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) = Relation Between Continuity & Differentiability | Mathematics (Maths) Class 12 - JEE, then find the value of k so that f(x) becomes continuous at x = 0. Hence, find all the points where the functions is non-differentiable.

Sol. From the graph of f(x) it is clear that for the function to be continuous only possible value of k is 1.

Relation Between Continuity & Differentiability | Mathematics (Maths) Class 12 - JEE

Points of non-differentiability are x = 0, ±1.

 

Ex.16 If f(x) = Relation Between Continuity & Differentiability | Mathematics (Maths) Class 12 - JEE where [.] denotes the greatest integer function.

Discuss the continuity and differentiability of f(x) in [0, 2).

Sol. Since 1 Relation Between Continuity & Differentiability | Mathematics (Maths) Class 12 - JEE x - 1 < 1 then [x2 - 2x] = [(x - 1)2 - 1] = [(x - 1)2] - 1 = 0 - 1 = -1

Relation Between Continuity & Differentiability | Mathematics (Maths) Class 12 - JEE

Graph of f(x) :

Relation Between Continuity & Differentiability | Mathematics (Maths) Class 12 - JEE

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 : Relation Between Continuity & Differentiability | Mathematics (Maths) Class 12 - JEE ⇒  Relation Between Continuity & Differentiability | Mathematics (Maths) Class 12 - JEE 

Relation Between Continuity & Differentiability | Mathematics (Maths) Class 12 - JEE

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) = Relation Between Continuity & Differentiability | Mathematics (Maths) Class 12 - JEE. If f '' (1) exist then find the value of a2 + b2 + c2.

Sol. For continuity at x = 1 we leave f (1) = 1 and f (1+)  = a + b + c

a + b + c = 1 ....(1)

Relation Between Continuity & Differentiability | Mathematics (Maths) Class 12 - JEE  for continuity of  f ' (x) at x = 1 f ' (1) = 3; f ' (1+) = 2a + b

hence 2a + b = 3 ....(2)

Relation Between Continuity & Differentiability | Mathematics (Maths) Class 12 - JEE   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  

Relation Between Continuity & Differentiability | Mathematics (Maths) Class 12 - JEE

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 Relation Between Continuity & Differentiability | Mathematics (Maths) Class 12 - JEE ⇒ sin-1 |sin x| = Relation Between Continuity & Differentiability | Mathematics (Maths) Class 12 - JEE

Also cos-1 |sin x| = Relation Between Continuity & Differentiability | Mathematics (Maths) Class 12 - JEE - sin-1 |sin x|

Relation Between Continuity & Differentiability | Mathematics (Maths) Class 12 - JEE

Relation Between Continuity & Differentiability | Mathematics (Maths) Class 12 - JEE

⇒ f(x) is not differentiable at  Relation Between Continuity & Differentiability | Mathematics (Maths) Class 12 - JEE

⇒ f(x) is not differentiable at  Relation Between Continuity & Differentiability | Mathematics (Maths) Class 12 - JEE

Ex.19 Find the interval of values of k for which the function f(x) = |x2 + (k - 1) |x| - k| is non differentiable at five points.

Sol. 

Relation Between Continuity & Differentiability | Mathematics (Maths) Class 12 - JEE

f(x) = |x2 + (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 .

 

The document Relation Between Continuity & Differentiability | Mathematics (Maths) Class 12 - JEE is a part of the JEE Course Mathematics (Maths) Class 12.
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FAQs on Relation Between Continuity & Differentiability - Mathematics (Maths) Class 12 - JEE

1. What is the definition of continuity in mathematics?
Ans. Continuity in mathematics refers to the property of a function where the function is uninterrupted and has no jumps or breaks. It means that the function can be drawn without lifting the pen from the paper.
2. Can a function be continuous but not differentiable?
Ans. Yes, it is possible for a function to be continuous but not differentiable. A function can be continuous if there are no breaks or jumps, but it may not have a well-defined derivative at some points. This occurs when the function has a sharp corner, vertical tangent, or a cusp at a particular point.
3. Is every differentiable function continuous?
Ans. Yes, every differentiable function is continuous. If a function is differentiable at a point, it implies that the function is also continuous at that point. This is because the existence of the derivative requires the function to be continuous in the neighborhood of that point.
4. How can we determine if a function is differentiable at a specific point?
Ans. To determine if a function is differentiable at a specific point, we need to check if the derivative exists at that point. The derivative exists if the left-hand derivative is equal to the right-hand derivative, and the function is continuous at that point. If these conditions are satisfied, the function is differentiable at that point.
5. Can a function be differentiable but not continuous?
Ans. No, a function cannot be differentiable but not continuous. If a function is differentiable at a point, it implies that the function is also continuous at that point. Differentiability requires the function to be continuous in the neighborhood of that point.
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