Continuity — definition, types of discontinuity (removable, jump, infinite): Visual walkthrough

4.1.7 · D2 · Maths › Calculus I — Limits & Derivatives › Continuity — definition, types of discontinuity (removable,

Step 1 — Crayon test: "pen uthaye bina" ka matlab kya hai

KYA. Hum sabse bachpane wale idea se shuru karte hain: ek curve continuous hai agar tum use crayon se trace kar sako bina crayon ko kaagaz se uthaye. Figure mein do curves dekho — baayein wali ek hi stroke mein trace ho jaati hai, daayein wali tumhe pen uthane par majboor karti hai.

YAHAN SE KYU shuru karein. Jo bhi formula hum baad mein likhen, use yeh ek physical fact encode karna hoga. Agar koi formula "kya crayon utha?" se kabhi disagree kare, toh formula galat hai, crayon nahi.

PICTURE.

Figure — Continuity — definition, types of discontinuity (removable, jump, infinite)

Dhyaan se dekho woh exact jagah jahan pen daayein curve par uthta hai. Us spot ki ek horizontal position hai — use $a$ kehte hain (bas ek naam "x-coordinate jahan problem hai" ke liye). Ab se sab kuch $x = a$ par focus kiya hua microscope hai.

Step 2 — $a$ tak "pahunchne" ke do tarike: the one-sided limits

KYA. Lifting pen ko pakadne ke liye hum do cheezein compare karte hain: woh height jis taraf curve ja raha hai jab hum slide karte hain, aur woh height jo uski actually $a$ par hai. Lekin "ja raha hai" ke do directions hain — left se aur right se. Hum unhe alag track karte hain.

ALAG KYU. Kyunki jump ka poora drama yahi hai ki do directions alag-alag heights ki taraf jaate hain. Agar hum sirf ek direction dekhein toh aadhi kahani miss ho jaayegi.

PICTURE.

Do chhote bugs $x = a$ ki taraf chal rahe hain:

The cyan bug left se approach karta hai ( $x < a$ ). Jis height par woh converge karta hai woh likha jaata hai $L^{-} = lim_{x \to a^{-}} f (x)$ Term by term padhein: $lim$ = "woh value jis taraf ja rahe hain", $x \to a^{-}$ = "jaise $x$ neeche se $a$ ki taraf slide karta hai (chhota minus left side matlab hai)", $f (x)$ = "woh height jo hum dekh rahe hain".
The amber bug right se approach karta hai ( $x > a$ ), deta hai $L^{+} = lim_{x \to a^{+}} f (x)$ jahan $x \to a^{+}$ = "upar se $a$ ki taraf slide karna".

Step 3 — Teen heights jo agree karni chahiye

KYA. Ab hamare paas position $a$ par teen heights hain:

$L^{-}$ — jahan left neighbours point karte hain,
$L^{+}$ — jahan right neighbours point karte hain,
$f (a)$ — crayon ka actual dot.

Pen tab nahi uthta jab teenon same number hon aur woh number finite ho.

TEENON KYU. Har disagreement pen ke uthne ke ek alag tarike ko represent karti hai, aur yeh independently fail ho sakte hain — isliye hume sach mein teenon check karni padti hain.

PICTURE.

Figure mein teenon heights horizontal ticks ki tarah stack ki gayi hain. Jab yeh line up hoti hain (left panel) toh crayon smoothly glide karta hai. Jab koi bhi tick line se bahar ho (right panel) toh crayon jump karta hai.

Step 4 — Failure mode A: the hole (removable)

KYA. Maano dono trends agree karte hain aur same finite number ke barabar hain, jise hum ek chhota naam dete hain: let $L := L^{-} = L^{+}$ (toh $L$ simply "common height jo dono bugs report karte hain" hai). Saare neighbours is height $L$ ki taraf point karte hain — lekin dot $f (a)$ missing hai, ya galat height par baitha hai.

REMOVABLE KYU KEHTE HAIN. Neighbours already agree kar chuke hain ki dot kahan hona chahiye (yaani $L$ par). Hum defect ko ek single dot ko height $L$ par rakh ke (ya move karke) remove kar sakte hain. Ek dot sab fix kar deta hai.

PICTURE.

Curve dono taraf se smoothly height $L$ ki taraf sweep karta hai; bas $x = a$ par ek open circle (empty dot) hai, ya upar float karta ek stray filled dot.

Step 5 — Failure mode B: the step (jump)

KYA. Ab dono trends bilkul theek finite numbers hain, lekin woh disagree karte hain: $L^{-} \neq = L^{+}$ . Left neighbours ek height ki taraf point karte hain, right waale doosri height ki taraf.

EK DOT FIX KYU NAHI KAR SAKTA. Ek dot ki ek height hoti hai. Woh left trend ko match kar sakta hai ya right trend ko, dono ko kabhi nahi. Isliye koi bhi redefinition jump ko heal nahi kar sakti — gap dono alag-alag pieces mein baked in hai.

PICTURE.

Do curve segments do alag heights par land karte dikhte hain, unke beech ek vertical amber gap hai — jump size $jump = ∣ L^{+} - L^{-} ∣,$ jahan bars $∣ \cdot ∣$ ka matlab hai "distance, hamesha positive" (gap ka koi sign nahi, sirf size hoti hai).

Step 6 — Failure mode C: the pole (infinite)

KYA. Ek trend bilkul finite number nahi hoti: jaise bug $a$ ki taraf approach karta hai, height $+ \infty$ ki taraf rocket karti hai ya $- \infty$ ki taraf plunge karti hai. Curve kabhi land nahi karta. Do flavours hain:

Opposite signs — ek side $- \infty$ ki taraf jaati hai, doosri $+ \infty$ ki taraf (jaise $\frac{1}{x - 3}$ ).
Same sign — dono sides same tarike se blow up karti hain, e.g. dono $+ \infty$ ki taraf (jaise $\frac{1}{x ^{2}}$ at $0$ ). Kisi bhi case mein, kam se kam ek side infinite hai, isliye point ek infinite discontinuity hai.

YEH FUNDAMENTALLY ALAG KYU HAI. Hole aur step cases mein heights real numbers thi jinhe hum compare kar sakte the. Yahan kam se kam ek side ki koi finite height nahi — match karne ke liye kuch hai hi nahi, isliye yeh unfixable hai aur structurally sabse worst hai. Note karo ki "same sign" case bhi removable nahi hai: $+ \infty$ ki taraf jaata curve koi finite value nahi le sakta jise dot fill kar sake.

PICTURE.

Left panel: opposite signs (curve ek side par dive karta hai, doosri par soar). Right panel: same sign (dono sides $+ \infty$ ki taraf soar karti hain), classic $\frac{1}{x ^{2}}$ spike.

Step 7 — Subtle case: limit exist karta hai lekin dot misplaced hai

KYA. Yeh ek removable case hai jo laagta hai jaise continuous hona chahiye. Dono trends agree karte hain aur finite hain, dot exist karta hai — lekin dot galat height par baitha hai.

PHIR BHI DISCONTINUOUS KYU KEHTE HAIN. Continuity demand karti hai trend $=$ dot. Ek present-but-wrong dot phir bhi equality todta hai, isliye crayon phir bhi uthta hai (stray dot ki taraf) aur waapis aata hai.

PICTURE.

Smooth curve height $1$ ki taraf jaata hai; ek amber dot height $5$ par upar float kar raha hai, disconnected.

Step 8 — Failure mode D: the blur (essential / oscillatory)

KYA. Ab tak har bug ya toh ek height par settle hota tha (finite) ya infinity ki taraf escape karta tha. Ek aur possibility hai jo humne abhi nahi dikhaayi: ek bug jo kabhi settle hi nahi karta — height upar-neeche swinging karti rehti hai hamesha ke liye, teji se aur teji se, jaise bug $a$ ke paas aata hai. Na $L^{-}$ exist karta hai na $L^{+}$ (finite nahi, $\pm \infty$ bhi nahi). Yeh oscillatory kind ki essential discontinuity hai.

YEH CHAUTHA, ALAG CASE KYU HAI. Removable, jump aur infinite sab ne assume kiya tha ki one-sided limits ka kam se kam koi answer hota hai (finite ya $\pm \infty$ ). Yahan woh basic sawaal "bug kis height ki taraf ja raha hai?" ka koi jawab hi nahi — trend ek blur hai. Koi dot, aur koi infinity, use capture nahi kar sakti, isliye yeh sabse incurable hai.

PICTURE.

$x = 0$ ke paas $sin (1/ x)$ ka curve dekho: wiggles infinitely tight $y$ -axis ke khilaf pile up hote hain, $- 1$ aur $+ 1$ ke beech ki poori band ko sweep karte hain bina kabhi kisi value par home kiye.

Step 9 — Edge case: domain endpoint par continuity

KYA. Upar sab kuch assume karta tha ki $a$ ke dono sides par neighbours hain. Lekin bahut si functions line ke sirf ek part par rehti hain — e.g. $f (x) = x$ sirf $x \geq 0$ ke liye exist karta hai, isliye $a = 0$ uske domain ka left edge hai: left side se koi bug bhejne ke liye koi "left side" hi nahi hai.

RULE KO RELAX KYU KARNA PADEGA. Agar left mein koi territory hi nahi hai toh hum $L^{-}$ demand nahi kar sakte — sawaal wahan meaningless hai. Isliye endpoint par hum missing side drop karte hain aur sirf woh side rakhte hain jo domain ke andar hai.

PICTURE.

Sirf ek bug (domain ke andar se) endpoint ki taraf approach karta hai; doosri side "off the map" ki tarah greyed out hai.

Intuition Sabse extreme edge case: ek isolated point

Endpoint idea ko uski limit tak push karo. Maano $a$ ke koi bhi neighbours nahi hain kisi bhi side par — domain mein $a$ hai lekin uske paas kuch nahi (maano $f$ sirf single point ${5}$ par define hai, ya domain ${5} \cup [6, 7]$ jaisi hai jahan $5$ akela khada hai). Tab koi bug kisi bhi direction se bhejne ke liye nahi hai, isliye limit condition un neighbours ke baare mein sawaal poochti hai jo exist hi nahi karte.

Convention ise cleanly resolve karta hai: ek function trivially (vacuously) ek isolated point par continuous hai. Koi neighbours nahi jo dot se disagree karein, toh crayon kabhi pakda nahi ja sakta uthte hue — simply koi jagah hi nahi hai jahan se approach karein. Isliye isolated points hamesha continuous hote hain, $f (a)$ ki value chahe kuch bhi ho.

Ek-picture summary

Ek frame, paanch panels: glide (continuous), hole (removable), step (jump), pole (infinite), blur (essential/oscillatory). Left se right padho — yeh ek nazar mein poori classification hai, usi $L^{-}, L^{+}, f (a)$ ticks se keyed jo humne Step 3 mein banaye, saath mein endpoint/isolated-point reminders ki interior rules ko dono sides chahiye jabki lone points trivially continuous hain.

Recall Feynman retelling — poora walk simple words mein

Socho ek bug curve ke saath $a$ spot ki taraf crawl kar raha hai. Actually do bugs: ek left se aata hai, ek right se. Har bug woh height report karta hai jis taraf woh ja raha hai — woh reports hain $L^{-}$ aur $L^{+}$ . Alag se, hum $a$ par painted actual dot dekhte hain: woh hai $f (a)$ .

Agar dono bug-reports aur dot same finite number hain, crayon smoothly glide karta hai — continuous. Agar bugs aapas mein agree karte hain (ek height $L$ par) lekin dot missing hai ya galat jagah float kar raha hai, toh yeh ek hole hai jise hum ek dot se bhar sakte hain — removable. Agar do bugs alag finite heights report karte hain, toh yeh ek step hai jise koi single dot bridge nahi kar sakta — jump. Agar kisi bhi bug ki report kabhi settle nahi hoti — woh infinity ki taraf ek vertical wall ke saath screams off karti hai — toh yeh ek pole hai, ek infinite discontinuity (chahe do sides same taraf jaayen, dono $+ \infty$ ki taraf jaise $1/ x^{2}$ , ya opposite taraf jaise $1/ (x - 3)$ ). Aur agar ek bug ki height hamesha wiggle karti rehti hai bina kabhi koi value choose kiye, toh yeh ek blur hai — ek essential, oscillatory discontinuity, sabse hopeless.

Do twists edges par: function jahan rehti hai uske bilkul kinare par, tumhare paas bhejne ke liye sirf ek bug hota hai — toh tum sirf us ek side se poochho ki dot se match kare. Aur ek bilkul isolated point par jahan koi neighbours hi nahi hain, tumhare paas koi bug nahi hai — toh kuch crayon ko uthte pakad nahi sakta, aur point automatically continuous hai.

Toh: glide, hole, step, pole, ya blur — yeh hain woh tarike jaise tumhara crayon behave karta hai, edges aur lone points ko jinke bugs actually exist karte hain unse handle kiya jaata hai.

Recall Fast self-check

Kaunsa failure hai jahan $L^{-} = L^{+}$ finite hai lekin $f (a)$ wrong ya missing hai? ::: Removable (hole). Kaunsa case hai jahan $L^{-}, L^{+}$ finite hain lekin unequal hain? ::: Jump (step). Kaunsa case hai jahan ek side $\pm \infty$ ke barabar hai? ::: Infinite (pole / vertical asymptote). Kya dono sides SAME infinity ki taraf ja sakti hain aur phir bhi infinite ho sakti hain (removable nahi)? ::: Haan — e.g. $1/ x^{2}$ ki dono sides $\to + \infty$ jaati hain; koi finite dot use fix nahi karta. Kaunsa case hai jahan ek side oscillate karti hai aur kabhi settle nahi hoti (even $\pm \infty$ nahi)? ::: Essential / oscillatory (blur). Jump ko ek dot se patch kyun nahi kiya ja sakta? ::: Ek dot ki ek height hoti hai; woh ek saath do alag trends ke equal nahi ho sakti. Left domain endpoint par continuity ke liye kya hold karna chahiye? ::: Sirf $lim_{x \to a^{+}} f (x) = f (a)$ — missing left side required nahi hai. Kya ek function apne domain ke isolated point par continuous hoti hai? ::: Haan — trivially (vacuously); dot se disagree karne ke liye koi neighbours nahi hain.

Connections

Continuity — definition, types of discontinuity (removable, jump, infinite) (parent)
Limits — definition and one-sided limits (bugs hain one-sided limits)
Standard limit sin(x)/x = 1 (Step 7)
Piecewise functions (Step 5)
Vertical asymptotes and rational functions (Step 6)
Differentiability implies continuity (continuity derivatives ke liye entry ticket hai)
Intermediate Value Theorem (poore interval mein continuity chahiye, endpoints bhi)