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Continuity and Differentiability is one of the most important topics which forms the base to understand, continuity at a certain point, derivative of functions, and continuity on a given interval.
- A function's continuity specifies its properties and the value it serves as a function.
- If the derivative of the function f'(a) exists at every point in its specified domain, then the function f(x) is said to be differentiable at the point x = a.
- It can be used to illustrate differentiability.
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Key Terms: Continuity, Differentiability, Functions, Limits, Domain, Derivatives, Curves
Continuity
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A function's continuity specifies its properties and the value it serves as a function.If a curve is continuous at every point inside its domain and does not have any missing points or breaking points, the function is said to be continuous in nature.
- If all three of the given conditions are met, then a function f(x) is said to be a continuous function at the point x = an in its domain.
- The three conditions are mentioned below:
- The value of f(a) must be finite
- If Lim x→an f (x) is existing, is it essential that the right hand limit and left hand limit should be equal to each other, and both of them should be finite
- Then Lim x→an f (x) = f (a)
Thus, a function f(x) is said to be continuous at all positions in the range [a, b], including the endpoints a and b.
- Continuity of f at a is: Llim x→a+ f(x) = f (a)
- Continuity of f at b is: Lim x→b- f(x) = f (b)
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Geometrical Interpretation of Continuity
If there is no break in the function's graph at the point (c, f(c)), then the function f will be continuous at x = c. Similarly, if there is no break in the function's graph over the whole interval, the interval is said to contain a continuous function.
Geometrical Interpretation of Continuity
Discontinuity of Function
A function is discontinuous if the following situations are present:
- The value of f(a) is infinite
- If Lim x→an f (x) is existing, and the right hand limit and left hand limit are not equal to each other, and both of them are infinite
Differentiability
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In case of differentiability, if the derivative of the function f'(a) exists at every point within its defined domain, then the function f(x) is said to be differentiable at the point x = a.
The formula for differentiability is: f(a) =[f(a+h)−f(a)h]/h
Read More: Differentiation and Integration Formula
Theorems Associated With Continuity and Differentiability
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There are six theorems associated with Continuity And Differentiability, they are explained below:
Theorem 1: Continuous Function Algebra
If, for example, a and b, two real functions, which are continuous at real number c, then:
- a + b is continuous at x=c.
- a – b is continuous at x=c.
- a X b is continuous at x=c.
- a/b is continuous at x=c, (if g(c) ≠ 0).
Theorem 2
The second theorem states that if a and b are real-valued functions, then (a o b) is defined at c. If both g and f are continuous at c, then (a o b) is continuous at c as well.
Theorem 3
A function a is continuous at a point c if and only if it is differentiable at that location.
Theorem 4 (Chain Rule)
Assume that f is a real-valued function that combines the operations of y and z, with the formula f = y o z. Assuming t = y(x), we have df/dx = (dy/dt) if both dt/dx and dy/dt exist. (dt/dx)
Theorem 5
The theorem states that:
- d/dx(ex) = ex is the derivative of ex with respect to x.
- d/dx(log x) = 1/x is the derivative of log x with respect to x.
Theorem 6 (Rolle's Theorem)
Assuming that f: [a, b] → R is continuous on [a, b] and differentiable on (a, b), with a and b being some real numbers, then f(a) = f(b). After that, there is a c in (a, b) such that f'(c) = 0.
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| Relations and Functions | Types of Functions | |
Things to Remember
- When the function exists at point c the continuity of a real function (f) on a subset of the real numbers is defined, and represented as: Lim x→c f(x) = f(c)
- If a real function (f) is continuous throughout its whole domain, then it is said to be continuous.
- The function f(x) is said to be differentiable at the point x = a
- The situation of differentiability if the derivative of the function f'(a) exists at each point within its defined domain.
- To define the continuity of real functions, limits are frequently used.
- A function f with a variable named x is continuous at the real number f(c) if and only if the limit of f(x), as x approaches c, is equal to f.
Read More: Differential Equation
Previous Year Questions
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Sample Questions
Ques. What is the continuity of a function? (3 Marks)
Ans. The features of a function and the purpose it performs as a function are described by its continuity. A function is said to be continuous in nature if a curve is continuous at every point within its domain and lacks any missing points or breaking points. A function f(x) is said to be a continuous function at the point x = an in its domain if all three of the above conditions are satisfied. The following three circumstances are listed:
- There must be a finite value for f(a).
- Is it necessary for the right hand limit and left hand limit to be equal to each other and both of them to be finite if a Lim xan f (x) exists?
- Consequently, Lim xan f (x) = f (a).
Ques. Explain the continuity of the function f(a) = sin a . cos a (2 Marks)
Ans. The functions sin and cos are continuous. It is common knowledge that a continuous function also exists when two continuous functions are combined. As a result, the continuous function f(a) = sin a. cos a.
Ques. Given that g(b) = 1/b-1, identify the places at which the composite function y = g[g(b)] discontinues. (2 Marks)
Ans. Given that, g(b) = 1/b-1
We know that the function g(b) = 1/b-1 is discontinuous at b = 1
Now, for b ≠1,
- g[g(b)]= g(1/b-1)
- = 1/[(1/b-1)-1]
- = b-1/ 2-b, which is discontinuous at the point b = 2.
Therefore, the points of discontinuity are b = 1 and b=2.
Ques. What is meant by real functions? (3 Marks)
Ans. A real function, or a function from real numbers to real numbers, can be represented by a graph in the Cartesian plane; this function is continuous if the graph is a single unbroken curve with the full real line as its domain. To define the continuity of real functions, limits are frequently used. A function f with a variable named x is continuous at the real number f(c) if and only if the limit of f(x), as x approaches c, is equal to f.
Ques. If g (a) = |cos a|, find g’(3π/4) (3 Marks)
Ans. Since, g(a) = |cos a|
When π/2 <a< π, cos a < 0,
Thus, |cos a| = -cos a
It means that, g(a) = -cos a
Hence, g’(a) = sin a
Therefore, g’(3π/4) = sin (3π/4) = 1/√2
g’(3π/4) = 1/√2
Ques. What is differentiability? (2 Marks)
Ans. The function f(x) is said to be differentiable at the point x = an in the situation of differentiability if the derivative of the function f'(a) exists at each point within its defined domain.
Ques. Verify the mean value theorem for the following function g (v) = (v – 3) (v – 6) (v – 9) in [3, 5] (3 Marks)
Ans. g(v)=(v−3)(v−6)(v−9)
=(v−3)(v2−15v+54)
=v3−18v2+99v−162
f(b)∈(3,5)
f′(b)=f(5)−f(3)/5−3
f(5)=(5−3)(5−6)(5−9)
=2(−1)(−4)=−8
f(3)=(3−3)(3−6)(3−9)=0
f′(b)=8−0/2=4
∴f′(b)=3b2−36b+99
3b2−36b+99=4
3b2−36b+95=0
av2+cv+b=0
a=3 , c=−36 , b=95
b=36±√(36)2−4(3)(95)/2(3)
=36±√1296−1140/6
=36±12.496
b=8.8&b=4.8
b∈(3,5)
f(v)=(v−3)(v−6)(v−9) on [3,5]
Ques. Explain why the function g = |a| is continuous at a = 0. (3 Marks)
Ans. From the given function,
g(a) = {-a, if a<0 and a, if a≥0
Lima→0- g(a)= lima→0- (-a) = 0
Similarly for the right hand side,
Lima→0+ g (a)= lima→0+ (a) = 0
Therefore, for the both left hand and the right hand limit, the value of the function coincide at the point a = 0.
Therefore, g is continuous at the point a =0.
Ques. Show that d2 = cos z / (1-sin z)2 if y = tan z + sec z. (3 Marks)
Ans. y= tan z + sec z
Differentiate wih respect to z, we get
dy/dz = sec2 z + sec z tan z
= (1/ cos2 z) + (sin z/ cos2 z)
= (1+sinz)/ (1+sinz)(1-sin z)
dy/dz = 1/(1-sin z)
Now, again differentiate with respect to z, we will get
d2y / dx2 = -(-cosz )/(1- sin z)2
d2y / dx2 = cos z / (1-sinz)2.
Ques. What is the derivative of : d/dx (sinx), d/dx (cosx), d/dx (tanx) (3 Marks)
Ans. The derivative are as follows:
d/dx (sinx)= cos x
d/dx (cosx)= -sin x
d/dx (tanx)= sec2x
Ques. What is Roll's theorem? (2 Marks)
Ans. Roll's theorem states that, assuming that f: [a, b] → R is continuous on [a, b] and differentiable on (a, b), with a and b being some real numbers, then f(a) = f(b). then, there is a c in (a, b) such that f'(c) = 0.
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