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Continuity formulas

Master Continuity through 26 JEE Advanced-level formulas, systematically structured with every variable spelled out. Revise concept-wise, identify the areas where you need improvement, and focus your preparation with greater precision.

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Continuity, every formula

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Continuity at a point
$x \to a lim f (x) = f (a)$
Q1MCQAt a point
$f$ is continuous at $x=a$ iff:
- A
  $\lim_{x\to a}f(x)=f(a)$
- B
  $f(a)$ exists
- C
  $\lim_{x\to a}f$ exists
- D
  $f'(a)$ exists
Three conditions
$f (a) defined, x \to a lim f exists, x \to a lim f = f (a)$
Q1MCQThree conditions
Continuity at $a$ requires all of the following EXCEPT:
- A
  $f$ is differentiable at $a$
- B
  $f(a)$ is defined
- C
  $\lim_{x\to a}f$ exists
- D
  $\lim_{x\to a}f=f(a)$
Left continuity
$x \to a^{-} lim f (x) = f (a)$
Q1MCQLeft continuity
$f$ is left-continuous at $a$ if:
- A
  $\lim_{x\to a^{-}}f=f(a)$
- B
  $\lim_{x\to a^{+}}f=f(a)$
- C
  $f'(a)$ exists
- D
  $f(a)=0$
Right continuity
$x \to a^{+} lim f (x) = f (a)$
Q1MCQRight continuity
$f$ is right-continuous at $a$ if:
- A
  $\lim_{x\to a^{+}}f=f(a)$
- B
  $\lim_{x\to a^{-}}f=f(a)$
- C
  $f$ is bounded
- D
  $f(a)=1$
Continuity on [a,b]
$cts on (a, b), right-cts at a, left-cts at b$
Q1MCQOn [a,b]
Continuity on $[a,b]$ requires continuity on $(a,b)$ plus:
- A
  right-cts at $a$ and left-cts at $b$
- B
  differentiability at $a,b$
- C
  $f(a)=f(b)$
- D
  nothing more
Sum/difference continuous
$f \pm g continuous at a$
if f,g continuous at a
Q1MCQSum continuous
If $f,g$ are continuous at $a$, then $f+g$ is:
- A
  continuous at $a$
- B
  discontinuous at $a$
- C
  undefined
- D
  differentiable at $a$
Product continuous
$f \cdot g continuous at a$
Q1MCQProduct continuous
If $f,g$ are continuous at $a$, then $fg$ is:
- A
  continuous at $a$
- B
  discontinuous
- C
  $0$
- D
  undefined
Scalar multiple continuous
$c f continuous at a, c \in R$
Q1MCQScalar multiple
If $f$ is continuous, then $5f$ is:
- A
  continuous
- B
  discontinuous
- C
  undefined
- D
  constant
Quotient continuous
$\frac{f}{g} continuous at a if g (a) \neq = 0$
Q1MCQQuotient
$\dfrac{f}{g}$ is continuous at $a$ provided:
- A
  $g(a)\neq0$
- B
  $f(a)\neq0$
- C
  $g(a)=0$
- D
  $f(a)=g(a)$
Composite continuous
$f cts at a, g cts at f (a) \Rightarrow g \circ f cts at a$
Q1MCQComposite
If $f$ is cts at $a$ and $g$ is cts at $f(a)$, then $g\circ f$ is:
- A
  continuous at $a$
- B
  discontinuous at $a$
- C
  undefined
- D
  constant
Discontinuity
$f not continuous at a \Rightarrow f discontinuous at a$
Q1MCQDiscontinuity
$f(x)=\dfrac1x$ at $x=0$ is:
- A
  discontinuous
- B
  continuous
- C
  differentiable
- D
  bounded
Removable discontinuity
$x \to a lim f exists but \neq = f (a) (or f (a) undefined)$
Q1MCQRemovable
A removable discontinuity at $a$ means:
- A
  $\lim_{x\to a}f$ exists but $\neq f(a)$ (or $f(a)$ undefined)
- B
  $\lim$ does not exist
- C
  a one-sided limit is $\infty$
- D
  the limit oscillates
Missing-point discontinuity
$x \to a lim f exists, f (a) undefined$
Q1MCQMissing point
A missing-point discontinuity has:
- A
  $\lim$ exists, $f(a)$ undefined
- B
  $\lim$ exists, $f(a)$ defined but different
- C
  $\lim=\infty$
- D
  $\lim$ oscillates
Isolated-point discontinuity
$x \to a lim f exists, f (a) defined, lim \neq = f (a)$
Q1MCQIsolated point
An isolated-point discontinuity has:
- A
  $\lim$ exists, $f(a)$ defined, $\lim\neq f(a)$
- B
  $f(a)$ undefined
- C
  $\lim=\infty$
- D
  $\lim$ oscillates
Jump (finite) discontinuity
$x \to a^{-} lim f, x \to a^{+} lim f finite but unequal$
Q1MCQJump discontinuity
A finite (jump) discontinuity has:
- A
  finite but unequal one-sided limits
- B
  equal one-sided limits
- C
  an infinite limit
- D
  no limit at all sides
Jump of discontinuity
$x \to a^{+} lim f - x \to a^{-} lim f$
Infinite discontinuity
$at least one one-sided limit = \pm \infty$
Q1MCQInfinite discontinuity
$f(x)=\dfrac{1}{x-2}$ at $x=2$ has:
- A
  infinite discontinuity
- B
  removable discontinuity
- C
  jump discontinuity
- D
  no discontinuity
Oscillatory discontinuity
$limit oscillates between finite values (e.g. sin \frac{1}{x})$
Q1MCQOscillatory
$f(x)=\sin\dfrac1x$ at $x=0$ has:
- A
  oscillatory discontinuity
- B
  removable discontinuity
- C
  jump discontinuity
- D
  no discontinuity
Continuity of elementary functions
$polynomials, sin, cos, e^{x}, lo g cts on their domains$
Q1MCQElementary functions
Which is continuous on all of $\mathbb{R}$?
- A
  $f(x)=x^{2}+\sin x$
- B
  $f(x)=\dfrac1x$
- C
  $f(x)=\tan x$
- D
  $f(x)=\log x$
Boundedness on [a,b]
$f cts on [a, b] \Rightarrow k \leq f (x) \leq m$
attains min k, max m
Q1MCQBoundedness
A continuous function on a closed interval $[a,b]$:
- A
  attains a maximum and a minimum
- B
  is unbounded
- C
  has no maximum
- D
  is always zero
Intermediate value theorem
$f cts on [a, b], y_{0} between f (a), f (b) \Rightarrow f (c) = y_{0}$
Q1MCQIVT
The Intermediate Value Theorem guarantees, for $y_0$ between $f(a)$ and $f(b)$:
- A
  some $c$ with $f(c)=y_0$
- B
  $f$ is monotonic
- C
  $f$ is differentiable
- D
  $f(a)=f(b)$
Root existence (IVT)
$f (a) f (b) < 0 \Rightarrow f (c) = 0 for some c \in (a, b)$
Q1MCQRoot existence
If $f$ is continuous on $[a,b]$ with $f(a)f(b)<0$, then:
- A
  $f$ has a root in $(a,b)$
- B
  $f$ has no root
- C
  $f$ is constant
- D
  $f(a)=f(b)$
Attains every value in [k,m]
$range of a continuous f on [a, b] is [k, m]$
Q1MCQRange on [a,b]
The range of a continuous $f$ on $[a,b]$ with min $k$, max $m$ is:
- A
  $[k,m]$
- B
  $\{k,m\}$
- C
  $(k,m)$
- D
  $\mathbb{R}$
|f| continuity
$f cts \Rightarrow ∣ f ∣ cts (converse false)$
Q1MCQ|f| continuity
If $f$ is continuous, then $|f|$ is:
- A
  continuous
- B
  discontinuous
- C
  undefined
- D
  constant
Redefining to remove discontinuity
$set f (a) = x \to a lim f (x)$
Q1MCQRedefining
To remove a removable discontinuity at $a$, redefine $f(a)$ as:
- A
  $\lim_{x\to a}f(x)$
- B
  $0$
- C
  $f(a-1)$
- D
  $\infty$
Single-point continuity
$a function may be continuous at only one point$
Q1MCQSingle-point continuity
A function continuous at exactly one point:
- A
  can exist
- B
  cannot exist
- C
  must be constant
- D
  must be linear

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Continuity formulas

Continuity, every formula

Continuity at a point

Three conditions

Left continuity

Right continuity

Continuity on [a,b]

Sum/difference continuous

Product continuous

Scalar multiple continuous

Quotient continuous

Composite continuous

Discontinuity

Removable discontinuity

Missing-point discontinuity

Isolated-point discontinuity

Jump (finite) discontinuity

Jump of discontinuity

Infinite discontinuity

Oscillatory discontinuity

Continuity of elementary functions

Boundedness on [a,b]

Intermediate value theorem

Root existence (IVT)

Attains every value in [k,m]

|f| continuity

Redefining to remove discontinuity

Single-point continuity

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Other ways to revise this chapter

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