Informative line

Definition Of Continuity

Learn definition of continuity calculus, infinite and jump discontinuity, practice one sided continuity for piecewise functions and Graph Reflection of Discontinuity.

Continuity at a Point

A function 'f' is continuous at \(x=a\), if

\(\lim\limits_{ x\to a^+}f(x)=\lim\limits_{ x\to a^–}f(x)=f(a)\)

\(i.e,\)  \(\lim\limits_{ x\to a}=f(a)\)   \((\lim\limits_{ x\to a}f(x)\,\text{will exist only if }\lim\limits_{ x\to a^+}f(x)=\lim\limits_{ x\to a^–}f(x)).\)

  • \(f(a)\) is defined as an assumption in the above definition.
  • If 'f' is not continuous at \(x=a\), we say it is discontinuous at \(x=a\).

Continuity at  \(x=a\)

\(\Rightarrow\,f(a)=f(a^+)=f(a^-)\)

 

(No break at \(x=a\))

Illustration Questions

A function 'f' is continuous at  \(x=3\), if  \(\lim\limits_{ x\to 3}f(x)=7\), then the value of \(f(3)\) is

A –9

B 11

C 17

D 7

×

If 'f' is continuous at \(x=3\), then  \(\lim\limits_{ x\to 3}f(x)=f(3)\) (by definition). 

 

\(\therefore\,f(3) =7\)

A function 'f' is continuous at  \(x=3\), if  \(\lim\limits_{ x\to 3}f(x)=7\), then the value of \(f(3)\) is

A

–9

.

B

11

C

17

D

7

Option D is Correct

Graphical Representation of Discontinuity

  • If a function \('f'\) is discontinuous (not continuous) at \(x=a\), then there will be a break in the graph at \(x=a\).

Illustration Questions

Consider the graph of a function \('f'\) as shown. The value of  \(x\) at which \('f'\) is discontinuous is - 

A 5

B –2

C 4

D –7

×

The graph of  \('f'\) is broken at \(x=–2\). We have to remove the pen as we go below \((–2,\,0)\). So \(x=–2\) is a point of discontinuity. 

Consider the graph of a function \('f'\) as shown. The value of  \(x\) at which \('f'\) is discontinuous is - 

image
A

5

.

B

–2

C

4

D

–7

Option B is Correct

Removable Discontinuity

  • If a function \('f'\) is discontinuous at \(x=a\), due to the fact that limit at \(x=a\) exists but is not equal to \(f(a)\).

\(i.e,\)  \(\lim\limits_{x\to a^+}f(x)=\lim\limits_{x\to a–}f(x)\neq f(a)\)

Then, it is called removable discontinuity.

  • The name removable is given because it can be removed by redefining. 

\(f(a)=\lim\limits_{x\to a}f(x)\)

\(\lim\limits_{x\to a^+}f(x)=\lim\limits_{x\to a–}f(x)\ne\,f(a)\)

 

  • If we redefine \(f\) at \(x=a\) i.e \(f(a)=\lim\limits_{x\to a}f(x)\) the dot gets filled and graph becomes continuous.

Note : we have to evaluate the limit in such problem but the language is of continuity.

Illustration Questions

Let   \(f(x)=\dfrac{x^2+3x–4}{x–1}\) ,  \('f'\) is discontinuous at  \(x=1\), what should be the value of \(f(1)\) so that \('f'\) becomes continuous at \(x=1\)?

A –11

B 5

C 17

D –18

×

For removing discontinuity at \(x=1\), we redefine

\(f(1)=\lim\limits_{x\to 1}\dfrac{x^2+3x–4}{x–1}\)

\(=\lim\limits_{x\to 1}\dfrac{(x-1)(x+4)}{(x–1)}\)

\(=\lim\limits_{x\to 1}(x+4)=5\)

 

Let   \(f(x)=\dfrac{x^2+3x–4}{x–1}\) ,  \('f'\) is discontinuous at  \(x=1\), what should be the value of \(f(1)\) so that \('f'\) becomes continuous at \(x=1\)?

A

–11

.

B

5

C

17

D

–18

Option B is Correct

Infinite Discontinuity

  • If the discontinuity is due to the fact that  \(\lim\limits_{x\to a}f(x)=\infty\) (not a finite number) then, it is called infinite discontinuity.
  • \(f(x)=tan\,x\) has an infinite discontinuity at \(x=(2n+1)\dfrac{\pi}{2}\) where,  \(n\) is an integer.

\(f(x)=tan\,x\) at \(x=\dfrac{\pi}{2}\)

\(\lim\limits _{x\to{\pi}/{2}^+}tan\,x=–\infty\)  and  \(\lim\limits _{x\to{\pi}/{2}^–}tan\,x=\infty\)

At least one of two limits R.H.L or L.H.L is infinity.

Illustration Questions

Which of the following function has an infinite discontinuity at indicated point ?

A \(f(x)=\dfrac{1}{x+3}\) at \(x=–3\)

B \(f(x)=\dfrac{1}{x}\) at \(x=2\)

C \(f(x)=\dfrac{2}{x^2}\) at \(x=2\)

D \(f(x)=\dfrac{1}{x+7}\) at \(x=1\)

×

For option a,

\(\lim\limits_{x\to –3}\,\dfrac{1}{x+3}=\infty\), all the other three limit exists and have finite value (direct substitution method).

\(\therefore\) answer is (a)

For option 'b' \(\to\) \(\lim\limits_{x\to 2}\,\dfrac{1}{x}=\dfrac{1}{2}\)

Hence, this option is incorrect.

For option c\(\to\) \(\lim\limits_{x\to 2}\,\dfrac{2}{x^2}=\dfrac{1}{2}\),

Hence, this option is incorrect.

For option 'd'\(\to\) \(\lim\limits_{x\to 1}\,\dfrac{1}{x+7}=\dfrac{1}{8}\)

Hence, this option is incorrect.

 

Which of the following function has an infinite discontinuity at indicated point ?

A

\(f(x)=\dfrac{1}{x+3}\) at \(x=–3\)

.

B

\(f(x)=\dfrac{1}{x}\) at \(x=2\)

C

\(f(x)=\dfrac{2}{x^2}\) at \(x=2\)

D

\(f(x)=\dfrac{1}{x+7}\) at \(x=1\)

Option A is Correct

Jump Discontinuity

If the discontinuity at \(x=a\) is due to the fact that limit doesn't exist at \(x=a,\,i.e\)

\(\lim\limits_{x\to a^+}f(x)\ne\lim\limits_{x\to a^-}f(x)\), we say that \('f'\) has a jump discontinuity at \(x=a\).

  • Such a discontinuity is reflected by a jump in the graph of function at \(x=a\). (therefore the name).
  • \(f(x)=[[x]]\) has such a discontinuity at every integer.

Illustration Questions

Let, \(f(x)= \begin{cases} \dfrac{|x|}{x} & if & x\neq0 \\ 1 & if & x=0 \end{cases}\) then, which of the following is true ?

A \(f\) has a removable discontinuity at \(x=0\)

B \(f\) has a jump discontinuity at \(x=0\)

C \(f\) is continuous at \(x=0\)

D \(f\) has an infinite discontinuity at \(x=0\)

×

Consider at  \(x=0\),

R.H.L = \(\lim\limits_{x\to 0^+}f(x)\) (0> 0).

\(=\lim\limits_{x\to 0^+}\dfrac{|x|}{x}\)

\(=\lim\limits_{x\to 0^+}\dfrac{x}{x}\)

\(=\lim\limits_{x\to 0^+}1=1\)

L.H.L = \(\lim\limits_{x\to 0^-}f(x)\) (0 < 0)

\(=\lim\limits_{x\to 0^-}\dfrac{|x|}{x}\)

\(=\lim\limits_{x\to 0^-}\dfrac{x}{x}=-1\)

\(\therefore\) R.H.L \(\ne\) L.H.L

\(\therefore\) jump discontinuity at, \(x=0\)

Let, \(f(x)= \begin{cases} \dfrac{|x|}{x} & if & x\neq0 \\ 1 & if & x=0 \end{cases}\) then, which of the following is true ?

A

\(f\) has a removable discontinuity at \(x=0\)

.

B

\(f\) has a jump discontinuity at \(x=0\)

C

\(f\) is continuous at \(x=0\)

D

\(f\) has an infinite discontinuity at \(x=0\)

Option B is Correct

One Sided Continuity

  • A function \('f'\) is said to be continuous from the right or right continuous at a value \(x=a\) if

 \(\lim\limits_{x\to a^+}f(x)=f(a)\)

i.e  R.H.L = \(f(a)\). (L.H.L is a different value)

  • Similarly, it is said to be continuous from the left or left continuous at a value \(x=a\) if

\(\lim\limits_{x\to a^-}f(x)=f(a)\)

i.e  L.H.L = \(f(a)\), (R.H.L is a different value)

  • \(f(x)=[[x]]\) is continuous from right at every integral value of  \(x\)

Illustration Questions

Which of the following functions, whose graphs are shown, have only right continuity at \(x=2\) ?

A

B

C

D

×

In option 'a',

\(f(2)\) = R.H.L \(\ne\)L.H.L

\(\to\) so, it has right continuity at \(x=2\).

Which of the following functions, whose graphs are shown, have only right continuity at \(x=2\) ?

A image
B image
C image
D image

Option A is Correct

One sided continuity for piecewise functions

 

  • A function \('f'\) is said to be continuous from the right or right continuous at a value \(x=a\) if

 \(\lim\limits_{x\to a^+}f(x)=f(a)\)

i.e  R.H.L = \(f(a)\). (L.H.L is a different value)

  • Similarly, it is said to be continuous from the left or left continuous at a value \(x=a\) if

\(\lim\limits_{x\to a^-}f(x)=f(a)\)

i.e  L.H.L = \(f(a)\), (R.H.L is a different value) 

  • \(f(x)=[[x]]\) is continuous from right at every integral value of  \(x\)

  • If \('f'\) is a piecewise function then we can find the right and left hand limits at all point when the definition expression of f changes.
  • e.g

\(f(x)= \begin{cases} 2x+5 & if & x<1\\ x^2+4 & if & x \geq1\\ \end{cases}\)

At x=1,

\(\lim\limits_{x\to 1^+}(x^2+4)=5\)= R.H.L

\(\lim\limits_{x\to 1^-}(2x+5)=7\) = L.H.L

\(f(1)=1+4=5\)

\(\therefore\,f(1)=\) R.H.L \(\ne\) L.H.L

\(\therefore\,f\) is right continuous at \(x=1\).

Illustration Questions

Let \(f(x)= \begin{cases} 2+x^2 & if & x\leq1\\ 2-x & if & 1<x <2\\ x+3 & if & x\geq2 \end{cases}\) Then choose correct option about  \('f'\).

A \(f\) is right continuity at \(x=1\)

B \(f\) is left continuity at \(x=1\)

C \(f\) is continuous at \(x=1\)

D Nothing can be said about continuity of  \('f'\) at \(x=1\)

×

At \(x=1\)

R.H.L \(=\lim\limits_{x\to 1^+}f(x)=\lim\limits_{x\to 1^+}(2-x)=2-1=1\)

L.H.L\(=\lim\limits_{x\to 1^-}f(x)=\lim\limits_{x\to 1^-}(2+x^2)=2+1=3\)

\(f(1)=2+1=3\)

L.H.L\(=f(1)\Rightarrow\,\)Left continuity at \(x=1\)

Let \(f(x)= \begin{cases} 2+x^2 & if & x\leq1\\ 2-x & if & 1<x <2\\ x+3 & if & x\geq2 \end{cases}\) Then choose correct option about  \('f'\).

A

\(f\) is right continuity at \(x=1\)

.

B

\(f\) is left continuity at \(x=1\)

C

\(f\) is continuous at \(x=1\)

D

Nothing can be said about continuity of  \('f'\) at \(x=1\)

Option B is Correct

Graphical Difference for Various Discontinuities

Consider the graphs of functions having various types of discontinuities.

 

\('f'\) has a removable discontinuity at \(x=a\) in the graph shown.

 

\('f'\) has infinite discontinuity at \(x=a\) in the graph shown.

R.H.L \(\ne\) L.H.L (R.H.L – L.H.L) = Jump.

\(f\) has a jump discontinuity at \(x=a\), there is a jump in the graph at \(x=a\).

Illustration Questions

The graphs of four functions are given with discontinuity nature written with it. Choose incorrect option.

A

B

C

D

×

(1) Removable discontinuity \(\Rightarrow\) break but no jump.

(2) Jump discontinuity \(\Rightarrow\) jump at the point. R.H.L \(\neq\) L.H.L

(3) Infinite discontinuity \(\Rightarrow\) Infinite jump at \(x=a\)

(At least one of R.H.L, L.H.L is infinite)

\(\therefore\) option 'b' is incorrect.

The graphs of four functions are given with discontinuity nature written with it. Choose incorrect option.

A image
B image
C image
D image

Option B is Correct

Practice Now