NDA Maths · Limits & Continuity

Continuity & Differentiability

A function is continuous at a point when the left limit, the right limit, and the function's value all agree — and continuity is the necessary (not sufficient) condition for differentiability.

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Why this matters

Most continuity questions either ask you to fix a parameter so the pieces meet, or to classify a discontinuity. The recurring trap is the continuous-but-not-differentiable corner, and oscillatory functions like sin(1/x) that have no limit at all.

Concept 1 of 4: The definition of continuity

Continuous at

a

means the graph has no break there: the value you approach equals the value you land on. Three things must all be equal — the left limit, the right limit, and

f(a)

Definition

$f$ is continuous at $a$ iff $\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=f(a)$ (all three exist and are equal). Polynomials, $\sin$ , $\cos$ , $e^x$ are continuous everywhere; rational functions are continuous except where the denominator vanishes. A removable discontinuity (a 0/0 hole) is patched by defining $f(a)=\lim_{x\to a}f(x)$ .

Continuity test at a point

\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=f(a)

Worked example

f(x)=\begin{cases}\dfrac{x^2-1}{x-1},&x\neq 1\\ 2,&x=1\end{cases}

continuous at

x=1

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

NDA · Apr 2018 · Q80Easy

Example 1 · Limits & Continuity · Continuity and Differentiability — Piecewise, Modulus, Composed, Oscillatory

f(x)=\frac{x^2-9}{x^2-2x-3}

x\ne3

is continuous at x=3, then?

Continuity needs the limit to EQUAL the value

A function can have a perfectly good limit at

a

and still be discontinuous — if

\lim_{x\to a}f(x)\neq f(a)

. All three of LHL, RHL, and

f(a)

must coincide. "The limit exists" alone is not continuity.

Drill 6 more on the definition of continuity

Concept 2 of 4: Finding parameters so f is continuous

When a piecewise function carries unknown constants and is declared continuous, set the one-sided limits equal to each other and to

f

at every join. Each join gives one equation; solve the system for the unknowns.

Definition

At each join $x=c$ : impose $\lim_{x\to c^-}f=\lim_{x\to c^+}f=f(c)$ . With $k$ joins and $k$ unknowns you get $k$ equations — solve simultaneously. (Continuity needs only value-matching; differentiability would additionally need slope-matching.)

Worked example

Find

k

f(x)=\begin{cases}kx+1,&x\le 2\\ 3x-1,&x>2\end{cases}

is continuous.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

NDA · Sep 2019 · Q98Easy

Example 2 · Limits & Continuity · Continuity and Differentiability — Piecewise, Modulus, Composed, Oscillatory

For what value of

k

is the function

f(x) = \begin{cases} 2x + \dfrac{1}{4}, & x < 0 \\ k, & x = 0 \\ \left(x + \dfrac{1}{2}\right)^{2}, & x > 0 \end{cases}

continuous ?

Drill 10 more on finding parameters so f is continuous

Concept 3 of 4: Types of discontinuity (removable, jump, oscillatory)

Discontinuities come in flavours: a removable hole (the limit exists but ≠ f(a)), a jump (LHL and RHL both exist but differ), and an oscillatory/essential break where no limit exists — the classic being sin(1/x) near 0. Greatest-integer functions create jump discontinuities at every integer.

Definition

Removable: $\lim_{x\to a}f$ exists but $\neq f(a)$ (or $f(a)$ undefined) — patchable.
Jump: LHL $\neq$ RHL, both finite (e.g. $\lfloor x\rfloor$ at integers).
Oscillatory/essential: no limit — $\sin\tfrac1x$ and $\sin\tfrac{1}{x^2}$ oscillate infinitely near 0.

Greatest-integer-built functions like $\lfloor x\rfloor^2-\lfloor x^2\rfloor$ are discontinuous at integers.

Worked example

Classify the discontinuity of

f(x)=\sin\tfrac1x

x=0

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

NDA · Sep 2021 · Q78Moderate

Example 3 · Limits & Continuity · Continuity and Differentiability — Piecewise, Modulus, Composed, Oscillatory

Consider the following statements in respect of the function

f(x)=\sin\!\left(\frac{1}{x^{2}}\right)

x\ne0

: 1. It is continuous at

x=0

, if

f(0)=0

. 2. It is continuous at

x=\frac{2}{\sqrt{\pi}}

. Which of the above statements is/are correct?

Removable vs jump — the limit's existence is the divider

A removable discontinuity has a single limit (LHL

=

RHL) that just misses

f(a)

— it can be patched. A jump has LHL

\neq

RHL (both finite), which can never be patched.

\lfloor x\rfloor

at integers is a jump, not removable.

Drill 4 more on types of discontinuity (removable, jump, oscillatory)

Concept 4 of 4: Continuity vs differentiability

Differentiability is strictly stronger: differentiable ⇒ continuous, but not the reverse. The standard counter-examples are corners (

|x|

) and steps (

\lfloor x\rfloor

) — continuous (for the corner) yet not differentiable. Continuity is also preserved by sums, products, and composition of continuous functions.

Definition

Differentiable at $a$ ⇒ continuous at $a$ (not conversely). $|x|$ is continuous at 0 but not differentiable (corner).
Closure: if $f,g$ are continuous at $a$ , so are $f\pm g$ , $fg$ , $f\circ g$ , and $f/g$ (where $g(a)\neq0$ ).
A product can be continuous even when a factor is awkward (e.g. $x\sin\tfrac1x\to 0$ by the squeeze).

Worked example

f(x)=|x|

continuous and differentiable at

x=0

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

NDA · Apr 2026 · Q71Easy

Example 4 · Limits & Continuity · Continuity and Differentiability — Piecewise, Modulus, Composed, Oscillatory

f(x)

is differentiable at

x=a

, then consider the following statements: I.

f(x)

is continuous at

x=a

. II.

\lim_{x\to a}f(x)=f(a)

. Which of the statements given above is/are correct?

Drill 12 more on continuity vs differentiability

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

The definition of continuity
Continuity test at a point
$\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=f(a)$

Watch out for (2)

Continuity needs the limit to EQUAL the value
→ The definition of continuity
Removable vs jump — the limit's existence is the divider
→ Types of discontinuity (removable, jump, oscillatory)

Test yourself on Limits & Continuity

20 past NDA questions from this chapter, timed at 25 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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Related notes

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