Directional Continuity and Continuity over an Interval

Continuity – Definition and Concept

A real function $f$ is said to be continuous if it is continuous at every point in its domain. In simple terms, a function is continuous when its graph has no breaks or jumps.

Suppose $f$ is a function defined on a closed interval $[a, b]$. Then, for $f$ to be continuous, it must be continuous at every point within $[a, b]$, including the endpoints $a$ and $b$.

Continuity of $f$ at $a$ means
$\lim_{x \rightarrow a^{+}} f(x) = f(a)$

and continuity of $f$ at $b$ means
$\lim_{x \rightarrow b^{-}} f(x) = f(b)$

Note that $\lim_{x \rightarrow a^{-}} f(x)$ and $\lim_{x \rightarrow b^{+}} f(x)$ do not make sense for a closed interval. If $f$ is defined only at one point, it is still considered continuous there — that is, if the domain of $f$ has only one element, $f$ is a continuous function.

Geometrical Interpretation of Continuity

When the graph of a function breaks or shows a gap at a certain point, it indicates a discontinuity.

If $\lim_{x \rightarrow a^{-}} f(x) = \lim_{x \rightarrow a^{+}} f(x)$, the limit exists, but for the function to be continuous, this common value must also equal $f(a)$.

So, $f(x)$ is continuous at $x = a$ if:
$\lim_{x \rightarrow a^{-}} f(x) = \lim_{x \rightarrow a^{+}} f(x) = f(a)$

This means the left-hand limit (LHL), right-hand limit (RHL), and the function’s actual value are all equal at that point.

Continuity in a Closed Interval

A function $f(x)$ is said to be continuous in a closed interval $[a, b]$ if:

1. $f$ is continuous at each and every point in $(a, b)$

2. The right-hand limit at $x = a$ exists and
   $\lim_{x \rightarrow a^{+}} f(x) = f(a)$

3. The left-hand limit at $x = b$ exists and
   $\lim_{x \rightarrow b^{-}} f(x) = f(b)$

Hence, continuity can be described in two ways:

• Continuity at a point

• Continuity over an interval

Directional Continuity and Continuity over an Interval

This section covers the complete concept of directional continuity (left-hand and right-hand continuity) and continuity of functions over intervals (open and closed). You’ll understand how limits from both sides determine smoothness at a point, along with examples, formulas, and solved questions based on NCERT and JEE patterns.

Directional Continuity – Left and Right Continuity

A function $y = f(x)$ is left-continuous at $x = a$ if:
$\lim_{x \rightarrow a^{-}} f(x) = f(a)$
or equivalently,
$\lim_{h \rightarrow 0^{+}} f(a - h) = f(a)$

A function $y = f(x)$ is right-continuous at $x = a$ if:
$\lim_{x \rightarrow a^{+}} f(x) = f(a)$
or equivalently,
$\lim_{h \rightarrow 0^{+}} f(a + h) = f(a)$

A function is said to be directionally continuous at a point when its limit exists and matches the function’s value when approached from one or both directions.

Continuity over an Open Interval $(a, b)$

A function $f(x)$ is continuous over an open interval $(a, b)$ if it is continuous at every point in that interval.

For any $c \in (a, b)$, $f(x)$ is continuous if
$\lim_{x \rightarrow c^{-}} f(x) = \lim_{x \rightarrow c^{+}} f(x) = f(c)$

This means there are no breaks or jumps in the graph of $f$ between $a$ and $b$.

Continuity over a Closed Interval $[a, b]$

A function $f(x)$ is continuous over a closed interval $[a, b]$ if:

• It is continuous at every point in $(a, b)$

• It is right-continuous at $x = a$

• It is left-continuous at $x = b$

At $x = a$, we must check:
$f(a) = \lim_{x \rightarrow a^{+}} f(x) = \lim_{h \rightarrow 0^{+}} f(a + h)$

At $x = b$, we must check:
$f(b) = \lim_{x \rightarrow b^{-}} f(x) = \lim_{h \rightarrow 0^{+}} f(b - h)$

Thus, for closed intervals, only the one-sided limits (RHL at $a$, LHL at $b$) are considered.

Example: Prove that $f(x) = [x]$ (greatest integer function) is not continuous in $[2, 3]$.

Condition 1: Continuity in $(2, 3)$

At any point $x = c$ in $(2, 3)$,
$f(c) = [c] = 2$

$LHL$ at $x = c$: $\lim_{x \rightarrow c^{-}} [x] = 2$
$RHL$ at $x = c$: $\lim_{x \rightarrow c^{+}} [x] = 2$

Hence, $f(x)$ is continuous for every $c$ in $(2, 3)$.

Condition 2: Right Continuity at $x = 2$

$f(2) = 2$ and
$\lim_{x \rightarrow 2^{+}} [x] = \lim_{h \rightarrow 0^{+}} [2 + h] = 2$

Thus, $f(x)$ is right-continuous at $x = 2$.

Condition 3: Left Continuity at $x = 3$

$f(3) = 3$ and
$\lim_{x \rightarrow 3^{-}} [x] = \lim_{h \rightarrow 0^{+}} [3 - h] = 2$

Since $f(3) \neq LHL$, $f(x)$ is not left-continuous at $x = 3$.

Therefore, the third condition fails - hence $f(x)$ is not continuous in $[2, 3]$.

Solved Examples Based On Directional Continuity and Continuity over an Interval

Example 1: $f(x)=[x]$ is continuous at each point of which of the following intervals?

1) $(1,2)$

2) $(1,3)$

3) $(-1,1)$

4) $(\frac{1}{2},\frac{3}{2})$

Solution:

As we have learned

Continuity in an open interval -

$\mathrm{F}(\mathrm{x})$ is said to be continuous in an open interval ( $\mathrm{a}, \mathrm{b}$ ) or $\mathrm{a}<\mathrm{x}<\mathrm{b}$ if it is continuous at each and every point of the interval belonging to its domain.
$f(x)=[x]$ will be discontinuous at integers . In (B), (C), (D) there are integers, lying in the interval in (B), (C), and (D), $f(x)$ will be continuous at each point. But in (A) it is

Hence, the answer is the option 1.
Example 2: $f(x)=x^2$ is continuous at each point of which of the following intervals?
1) $(1,5)$
2) $(5,7)$
3) $(7,9)$
4) All of them

Solution:
As we have learned
Continuity in an open interval -
$F(x)$ is said to be continuous in an open interval ( $a, b$ ) or $a<x<b$. If it is continuous at each and every point of the interval belonging to its domain.

At every $\mathrm{x}, x^2$ will give $\mathrm{LHL}, \mathrm{RHL}$, and function value all three equal so continuous everywhere so in all intervals it will be continuous

Hence, the answer is the option 1

Example 3: Which of the following statements is false?
1) $f(x)=\sin x$ is left continuously at $x=\pi / 2$
2) $f(x)=[x]$ is left continuous at $x=2$
3) $f(x)=|x|$ is left continuous at $x=0$
4) $f(x)=\left[x^2\right]$ is left continuous at $x=0$

Solution: To check left continuity we need to find LHL and function value at the point $x=a$
(A) $\rightarrow L H L=1, f(\pi / 2)=1$;
$(B) \rightarrow L H L=1, f(2)=2$;
$(C) \rightarrow L H L=0, f(0)=0$
$(D) \rightarrow L H L=0, f(0)=0$
$f(x)=[x]$ is not left-continuous at $x=2$
Hence, the answer is the option 2.
Example 4: Which of the following statements is false? ([.]= G.I.F)
1) $f(x)=[x]$ is continuous from right at $x=2$
2) $f(x)=[\sin x]$ is continuous from right at $x=-\pi / 2$
3) $f(x)=[\sin x]$ is continuous from right at $x=\pi / 2$
4) $f(x)=x^2$ is continuous from right at $x=2$

Solution:

Continuity from Right -

The function $f(x)$ is said to be continuous from right at

$
x=a: \text { if } \lim _{x \rightarrow a^{+}} f(x)=f(a)
$

$\ln (\mathrm{A}),(\mathrm{B})$ and $(\mathrm{D}) \rightarrow \mathrm{RHL}=$ function value
so $(A),(B),(D)$ are true
but in $(\mathrm{C}) \rightarrow \mathrm{RHL}=0, \mathrm{f}(\pi / 2)=1$
$\Rightarrow R H L \neq f(\pi / 2)$
$f(x)=[\sin x]$ is not continuous from right at $x=\pi / 2$
Hence, the answer is the option 3.
Example 5 : Which of the following functions is not continuous at all $x$ being in the interval $[1,3]$ ?
1) $f(x)=x^2$
2) $f(x)=x^3$
3) $f(x)=\sin x$
4) $f(x)=[x]$

Solution:

As we have learned

Continuity from Right -

$f(x)$ is said to be continuous in a closed interval $[a, b]$ or $a \leq x \leq b$ if
1. $f$ is continuous at each and every point in ( $a, b$ )
2. Right hand limit at $x=a$ must exist and

$
\lim _{x \rightarrow a^{+}} f(x)=f(a)
$

3. Left hand limit at $x=b$ must exist and

$
\lim _{x \rightarrow b^{-}} f(x)=f(b)
$

(A),(B),(C) are the functions which are continuous at every point in $(1,3)$ and for continuity at $\lim _{\mathrm{x}=1} f(x)=f(1) \quad \lim _{x \rightarrow 1^{+}} f(x)=f(3)$ and $x \rightarrow 3^{-}$also holds true
so (A),(B),(C ) are continuous at every point of $[1,3]$
In (D), $\mathrm{f}(\mathrm{x})=[\mathrm{x}]$ which will be discontinuous at $\mathrm{x}=2$ and $\mathrm{x}=3$ both as $\lim _{x \rightarrow 2^{+}} f(x), \lim _{x \rightarrow 2^{-}} f(x)$ and $\mathrm{f}(2)$ are not all equal and $\lim _{x \rightarrow 3^{-}} f(x) \neq f(3)$
$\therefore$ discontinuous at $\mathrm{x}=2$ and $\mathrm{x}=3$

List of topics related to Directional Continuity and Continuity over an Interval

This section lists all the subtopics connected to directional continuity and interval continuity. You’ll get a clear overview of related ideas such as differentiability, composite functions, and rate measures - all essential for mastering Class 12 Maths Chapter 5 and competitive exams like JEE and CUET.

Differentiability and Existence of Derivative

Examining differentiability Using Graph of Function

Derivative as Rate Measure: Definition, Formula, Examples

Continuity of Composite Function

Continuity And Differentiability

Differentiability of Composite Function

NCERT Resources

This part compiles all NCERT study materials related to this chapter - including notes, textbook solutions, and exemplar problems - to make your preparation complete and exam-ready.

NCERT Class 12 Maths Notes for Chapter 5 - Continuity and Differentiability

NCERT Class 12 Maths Solutions for Chapter 5 - Continuity and Differentiability

NCERT Class 12 Maths Exemplar Solutions for Chapter 5 - Continuity and Differentiability

Practice Questions based on Directional Continuity and Continuity over an Interval

This section includes topic-wise practice MCQs designed to test your understanding of directional and interval continuity. Each quiz helps you strengthen application-based skills and improve exam readiness.

Directional Continuity And Continuity Over An Interval- Practice Question MCQ

We have shared below the links to practice questions on the related topics of Directional Continuity and Continuity over an Interval: