Which of the following must be true of a continuous function on (a, b)?

The function achieves its maximum on (a, b).

For all Cauchy Sequences sn on the set (a, b), f(sn) is also Cauchy.

Continuity in Interval | Open and Closed Intervals

The feature of continuity can be seen on a day-to-day basis. For instance, the human heart beats continuously even when the person is sleeping. A continuous function is one which can be drawn on a graph paper without lifting a pen or a pencil. In this article, students will learn about continuity in an interval.

In mathematical terms, the definition of continuity is as follows.

A function f(x) is said to be continuous at a point if the following conditions are met:

The function at that point exists being finite.
The left and right-hand limit of the function is present.
The limit Lim_x→a f(x) = f(a) where is the point

The function f (x) is declared to be continuous if the above three conditions are satisfied for every point on the interval. If any of the functions are not satisfied, then the function is discontinuous at that point.

Basic Concepts on Continuity

The following are the basic concepts of continuity:

Open and Closed Intervals

An interval is called open if it doesn’t include the endpoints. It is denoted by ( ). For instance, (1, 2) means greater than 1 and less than 2. A closed interval is one which includes all the limit points. It is denoted by [ ]. For example, [2, 5] means greater than or equal to 2 and less than or equal to 5. An open interval is termed a half-open interval if it consists of one of its endpoints. It is written as ( ]. (0, 2] means greater than 0 and less than or equal to 2 and [0, 2) is greater than or equal to 0 and less than 2.

If a, b ∈ R and a < b, the following is a representation of the open and closed intervals.

Open interval is indicated by (a, b) = {x : a < y < b}.
Closed interval is indicated by [a, b] = {x : a ≤ x ≤ b}.
[ a, b ) = {x : a ≤ x < b} is an open interval from a to b, including a but excluding b
( a, b ] = { x : a < x ≤ b } is an open interval from a to b, including b but excluding a
The mandatory condition for continuity of the function f at point x = a [considering a to be finite] is that lim_x→a^– f(x) and lim_x→a⁺ f(x) should exist and be equal to f (a).
The function f (x) is said to be continuous:

In an open interval: if it is continuous at every point in the interval
In the closed interval: if

f(x) is continuous in (a, b)
Lim_x→a⁺f (x) = f(a)
lim_x→a^– f(x) = f(a)

Weierstrass Approximation Theorem

If f is a continuous real-valued function on [a, b] and if any ε >0 is given, then there exists a polynomial p on [a, b] such that

|f(x)-P(x)|< ε for all x in [a,b]

In other words, a continuous function on a closed and bounded interval can be uniformly approximated on that interval by polynomials to any degree of accuracy.

Also, Read

Limits Continuity and Differentiability

Differentiation

Functions

Continuity in Interval Examples

Example 1: If the function

\(\begin{array}{l}f(x)={\{\begin{array}{rlrlrlrlrlrl}&\frac{k\cos x}{\pi-2x},\text{when }x\neq\frac\pi2\\[3pt]&3,\text{ when }x=\frac\pi2\end{array}}\end{array} \)

be continuous at x = π / 2, then find the values of k.

Solution:

f (π / 2) = 3.

Since f(x) is continuous at x = π / 2

⇒ lim_x→π/2(k cosx / π − 2x) = f (π / 2)

⇒ k / 2 = 3

⇒ k = 6

Example 2: If

\(\begin{array}{l}f(x)={\{\begin{array}{rlrlrlrlrlrl}&\frac{{x^2}-4x+3}{{x^2}-1},\text{ for }x\neq1\\[3pt]&\text{ }2,\text{ for }x=1\end{array}}\end{array} \)

check whether the function is continuous. Justify your answer.

Solution:

\(\begin{array}{l}f(x)={\{\begin{array}{rlrlrlrlrlrl}&\frac{{x^2}-4x+3}{{x^2}-1},\text{ for }x\neq1\\[3pt]&\text{ }2,\text{ for }x=1\end{array}}\end{array} \)

f (1) = 2, f (1+) = lim_x→1+(x²− 4x + 3) /(x²− 1) = lim_x→1+(x − 3) / (x + 1) = −1

f (1−) = lim_x→1−(x²− 4x + 3)/(x²− 1) = −1

⇒ f (1) ≠ f (1−)

Hence, the function is discontinuous at x = 1

Example 3: What is the value of f(0) in the function f(x) = [(27 − 2x)^1/3−3] / 9 − 3 (243 + 5x)^1/5,

(x ≠ 0) is continuous?

Solution:

Since f (x) is continuous at x = 0,

f (0) = lim_x→0f(x) = lim_x→0[(27 − 2x)^1/3−3] / 9 − 3 (243 + 5x)^1/5, (Form 0 / 0)

= lim_x→0 [1 / 3] (27−2x)^−2/3(−2) / [−3 / 5] (243 + 5x)^−4/5(5)

= 2

Example 4: Let

\(\begin{array}{l}f(x)=\begin{cases}
\frac{x-4}{\left| x-4 \right|}+a & \text{ if } x<4 \\
a+b & \text{ if } x= 4\\
\frac{x-4}{\left| x-4 \right|}+b & \text{ if } x>4
\end{cases}\end{array} \)

then f (x) is continuous at x = 4 when

A) a = 0, b = 0

B) a = 1, b = 1

C) a = −1, b = 1

D) a = 1, b = −1

Solution:

We have L.H.L.

\(\begin{array}{l}=\underset{x\to 4}{\mathop{\lim }}\,f(x) \\ =\underset{h\to 0}{\mathop{\lim }}\,f(4-h)\\ =\underset{h\to 0}{\mathop{\lim }}\,\frac{4-h-4}{\left| 4-h-4 \right|}+\alpha \\ =\underset{h\to 0}{\mathop{\lim }}\,\left( -\frac{h}{h}+\alpha \right)\\ =a-1\end{array} \)

R.H.L

\(\begin{array}{l}=\underset{x\to 4}{\mathop{\lim }}\,f(x)\\ =\underset{h\to 0}{\mathop{\lim }}\,f(4+h)\\ =\underset{h\to 0}{\mathop{\lim }}\,\frac{4+h-4}{\left| 4+h-4 \right|}+b=b+1 \end{array} \)

Therefore, f(4) = a + b

Since f(x) is continuous at x = 4,

\(\begin{array}{l}\underset{x\to {{4}^{-}}}{\mathop{\lim }}\,f(x)=f(4)=\underset{x\to {{4}^{+}}}{\mathop{\lim }}\,f(x) \end{array} \)

a – 1 = a + b = b + 1

b = -1 and a = 1

Therefore, Option D is the answer.

Example 5: Let

\(\begin{array}{l}f(x)=\underset{n\to \infty }{\mathop{\lim }}\,\frac{\log (2+x)-{{x}^{2n}}\sin x}{1+{{x}^{2n}}},\end{array} \)

then

\(\begin{array}{l}A) f\ \text{is continuous at} x = 1\\ B) \underset{x\to {{1}^{+}}}{\mathop{\lim }}\, f(x)=log3\\ C) \underset{x\to {{1}^{+}}}{\mathop{\lim }}\, f(x)=-\sin 1\\ D) \underset{x\to {{1}^{-}}}{\mathop{\lim }}\, f(x) \text{does not exist}\\\end{array} \)

Solution:

\(\begin{array}{l}\text{For}\ \left| x \right|<1,{{x}^{2n}}\to 0\ \text{as}\ n\to \infty \end{array} \)

and for |x| > 1,

\(\begin{array}{l}1/{{x}^{2n}}\to 0 \text{as}\ \ n\to \infty ,\\ \text{So}, \ f(x)=\left\{ \begin{matrix} \log (2+x)\, \\ \underset{\to \infty }{\mathop{\lim }}\,\frac{{{x}^{-2n}}\log (2+x)-sinx}{{{x}^{-2n}}+1}=-\sin x, \\ \frac{1}{2}[log(2+x)-sinx],\, \\ \end{matrix} \right.\,\,\,\begin{matrix} \left| x \right|<1 \\ \left| x \right|>1 \\ \left| x \right|=1 \\ \end{matrix}\end{array} \)

Thus,

\(\begin{array}{l}\underset{x\to {{1}^{+}}}{\mathop{\lim }}\,f(x)=\underset{x\to 1}{\mathop{\lim }}\,(-\sin x)=-sin\ 1\ \text{and} \\ \underset{x\to {{1}^{-}}}{\mathop{\lim }}\,f(x)=\underset{x\to 1}{\mathop{\lim }}\,\,\,log(2+x) = \log 3.\end{array} \)

Example 6: If

\(\begin{array}{l}f(x)=\left\{ \begin{matrix} x-1,\,\,\,\,\,\,\,x<0 \\ {{x}^{2}}-2x,\,\,x\ge 0 \\ \end{matrix} \right.,\end{array} \)

then

A) f(|x|) is discontinuous at x = 0

B) f(|x|) is differentiable at x = 0

C) |f(x)| is non-differentiable at x = 0, 2

D) |f(x)| is continuous at x = 0

Solution:

where |x| ≤ 0 is not possible. Thus neglecting, we get

\(\begin{array}{l}f(\left| x \right|)=\begin{matrix} {{\left| x \right|}^{2}}-2\left| x \right|,\,\,\left| x \right|\ge 0 \\ \end{matrix}\\ f(|x)=\left\{\begin{matrix} x^{2}+2x &x<0 \\ x^{2}-2x& x\geq 0 \end{matrix}\right. \\ \therefore \ f'(\left| x \right|)=\left\{ \begin{matrix} x+2x, & x<0 \\ x-2x & x\ge 0 \\ \end{matrix} \right.\end{array} \)

Clearly, |f(x)| is continuous at x = 0 but non-differentiable at x = 0.

\(\begin{array}{l}f(x)=\left\{ \begin{matrix} \left| x \right|-1,\, \\ {{\left| x \right|}^{2}}-2\left| x \right|, \\ \end{matrix} \right.\,\,\,\,\,\,\,\,\,\begin{matrix} \left| x \right|<0 \\ \left| x \right|\geq 0 \\ \end{matrix}\\ g(x)=|f(x_=)\left\{\begin{matrix} 1-x, &x<0 & \\ -x^{2}+2x,&0\leq x<2 & \\ x^{2}-2x,&x\geq 2 & \end{matrix}\right.\end{array} \)

Clearly, |f(x)| is discontinuous at x = 0 but non-differentiable at x = 2.

Also,

\(\begin{array}{l}g'(x)=\left\{ \begin{matrix} -1, & x<0 \\ 2x+2, & 0<x<2. \\ 2x-2, & x>2 \\ \end{matrix} \right. \left| f(x) \right|\end{array} \)

is non-differentiable at x = 0 and x = 2.

Therefore, Option C is the answer.

Frequently Asked Questions

What are the conditions for a function f(x) to be continuous at a point?

A function is said to be continuous at a point, if it is defined at that point, its limit must exist at that point, and the value of the function at that point is equal to the value of the limit at that point.

Is continuity necessary for differentiability?

Yes, any differentiable function must be continuous at every point in its domain.

State Weierstrass approximation theorem.

If f is a continuous real-valued function on [a,b] and if any ϵ >0 is given, then there exists a polynomial p on [a, b] such that |f(x)-P(x)|< ϵ for all x ∈ [a,b]. In other words, a continuous function on a closed and bounded interval can be uniformly approximated on that interval by polynomials to any degree of accuracy.

Is sin x a continuous function?

Yes, sin x is continuous for every real number.

Continuity in Interval

Basic Concepts on Continuity

Open and Closed Intervals

Weierstrass Approximation Theorem

Also, Read

Continuity in Interval Examples

Related Video:

Limits, Continuity and Differentiability – Important Topics

Limits, Continuity and Differentiability – Important Questions

Frequently Asked Questions

What are the conditions for a function f(x) to be continuous at a point?

Is continuity necessary for differentiability?

State Weierstrass approximation theorem.

Is sin x a continuous function?

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