Continuity and Differentiability of a Function: Theorem & Examples

Continuity and Differentiability is an important unit in class 12 mathematics from the perspective of both boards and other competitive exams. It provides in-depth knowledge about the basics of continuity, differentiability, and the relation between them.

• Continuity and Differentiability will determine the value of derivatives of various quantities.
• It solves mathematic concepts such as logarithmic functions, exponential functions.
• The function solves continuity at a point, continuity on an interval and derivative of a function.
• The results are complementary to a function.
• Continuity function gives a graphical representation of a function in the form of a continuous wave.
• Differentiability function derives value for each domain.
• It is defined for closed interval [a, b].
• Continuous at the point x = a, if Lim_x→af(x)=f(a).
• It is differntiable at the same point x = a, if Lim_x→af′(a)=(f(a+h)−f(a))/h

Read More: Application of Derivatives

Key Terms: Continuity and Differentiability, Fundamental Rules, Logarithmic Functions, Exponential Functions, Continuity, Differentiability, Real Number, Domain, Chain Rule

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What is Continuity and Differentiability?

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Continuity and Differentiability are complementary functions. The concepts can solve both geometrical and algebraic functions. First, a function needs to be proved continuous for a point x=a.

• To determine the differential value, imagine that a function has a continuous domain value.
• The graph of the function does not have an intermediate value.
• It does not break in between the values of a function.
• Launching a rocket is a real-life example of continuity and differentiability.
• Suppose you want to use digital video recording equipment to record your playing video. 
• The video comes out as a continuous function.

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Definition of Continuity

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The term 'continuity' is derived from the word continuous, which means an endless or unbroken path. Continuity of a function can be better understood if defined in terms of limits.

• A real function f(x) can be considered as continuous at a point say, x = a, if the function f(x) is approaching to a is f(a).

lim_{x -> a} f(x) = f(a)

• The function must be continuous throughout its path at every point in its domain.

Continuity Defination

Conditions for continuity of a function

For any function to be continuous, it must meet the following conditions:

• The function f(x) specified at x = a, is continuous only if f(a) belongs to real number.
• The limit of the function as x approaches a, exists.
• It must be equal to the function value at x = a.

Defination and Conditions of Continuity

Discover about the Chapter video:

Continuity and Differentiability Detailed Video Explanation:

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Geometrical Representation of Continuity

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A function f shall be continuous at x = a, only if there is no break in the graph of the function at the point (a, f(a)). In the case of an interval, a function is said to be continuous if there are zero intervals in the graph of the given function throughout the interval.

Geometrical Representation of Continuity

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Definition of Differentiability

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A function f(x) is differentiable at a point x = a, if f ' (a), i.e., the derivative of the function exists at each point of its domain.

• The differentiability of a function is represented as:

f ' (x) = f (x + h) – f(x) / h

• If a function f is continuous at any point, the same function is also differentiable at any point x = c in its domain. 

Read More: Non-Parametric Test

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Important Differentiability Formula

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Some of the important differentiability formula are as follows:

Algebric Functions

Some important algebric functions are as follows:

• d(u+v) / dx = du/dx + dv/dx
• d/dx(uv) = udv/dx + vdu/dx
• d/dx(u/v) = vdu/dx - udv/dx / v²

Trigonometric Functions

Some important trigonometric functions used in continuity and differentiability are as follows:

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Theorem on Continuity and Differentiability

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The important theorems on continuity and differentiability are as follows:

Theorem 1

Two function f(x) and g(x) are said to be continuous in real-value at point x =d, then we have

• f(x) + g(x) is continuous at the point x = d
• f(x) - g(x) is continuous at a point x = d
• g(x).g(x) is continuous at point x = d
• f(x)/g(x) is continuous at a point x = d, provided g(d) ≠ 0

Theorem 2

For functions f(x) and g(x) such that the composite function fog(x) will be defined at point x = d. If g(x) is continuous at point x = d and the function f(x) is continuous at function g(d), then function fog(x) is continuous at x = d.

Theorem 3

For a given function f(x) which is differentiable at point d then function is continuous at point d. This proves that differential function is continuous.

Theorem 4

Let f : [a,b], -> R be continuous on [a,b] and differentiable on (a, b), such that f(a) = f(b), where a & b are some real numbers. Then at least one point 'c' must exist in (a, b), such that f ' (c) = 0. It is also known as Rolle’s Theorem.

Theorem 5

The theorem is based on Mean Value / Lagrange Theorem. Let f : [a,b] -> R be a continuous function on [a,b] and differentiable on (a,b). Then there exists at least one point 'c' in (a,b) such that

f ' (c) = \(\frac{f (b) - f (a)}{b-a}\)

Theorem 6

This theorem is based on Chain rule. Chain rule is basically used to differentiate the composite of functions. Let us suppose a function f= vou, if t=u(x), and both dt/dx and dv/dt exists, then

df/dx = dv/dt. dt/dx

Read More: Logarithm Formula

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Things to Remember

• Continuity and Differentiability function y = f(x) is defined over a closed interval [a, b].
• To determine whether a function is continuous, draw a graph without lifting a pen. 
• It is used to determine the profit and loss value in a marketplace.
• The function works on determining the root of a function.
• The graph drawn for continuity and differentiability function is not sharp at the edges.

Read More: Methods of Integration

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Previous Years Questions

• Define f(x)f(x) as the product of two real functions f1(x)=x,x∈f1(x)=x,x∈ R, and f2(x)
• d2xdy2 equals :​
• ddx{cosec−1(1+x22x)} is equal to​
• ddx(tan−1(√x−√a1+√xa)), x, a>0, is​
• Derivative of the function f(x)=log5(log7x), x>7 is​
• Differential coefficient of √sec√xsecx is​
• Differential coefficient of tan−12x1−x2 with respect to sin−12x1+x2 will be​
• For a>0,t∈(0,π2), let x=√asin−1t and y=√acos−1t, Then 1+(dydx)2 equals :​
• If $f(x) = cosx \cdot cos \,2x \cdot cos \,4x \cdot cos \,8x \cdot cos \,16x$, then the value of $f '\left(\frac{\pi}{4}\right)$ is​
• If g is the inverse function of f and f′(x)=sinx, then g′(x) is​
• If sec(x2−2xx2+1)=y, then dydx is equal to​
• If x=acosθ,y=bsinθ, then d3ydx3 is equal to​
• If x=exp{tan−1(y−x2x2)}, then dydx equals​
• If x=sintcos2t and y=costsin2t, then at t=π4, the value of dydx is equal to :​
• If y=√(1+cos2θ1−cos2θ) , then dydθ at θ=3π4 is:​
• If y=2x⋅32x−1, then dydx is equal to​
• If y=ax2+b, then dydx at x=2 is equal to​