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

4.1.7 · D1 · Maths › Calculus I — Limits & Derivatives › Continuity — definition, types of discontinuity (removable,

Parent note Continuity padhne se pehle, tumhe uske har symbol par command chahiye. Neeche, har symbol ko teen cheezein milti hain: plain words, picture, aur topic ko yeh kyun chahiye. Yeh is order mein hain ki har ek sirf upar wale par lean karta hai.

1. Function $f$ aur $f(a)$ ka matlab kya hai

Picture. Input $x$ ke liye horizontal axis aur output ke liye vertical axis draw karo. Har input $x$ ke liye ek dot rakho height $f(x)$ par. $x$ ko sweep karo aur dots ek graph trace karte hain. Woh single dot jo $x=a$ ke exactly upar hai, height $f(a)$ par, woh hai "woh value jo function $a$ par actually leta hai."

Topic ko yeh kyun chahiye. Continuity ki condition 1 literally hai "$f(a)$ is defined." Agar machine input $a$ par choke karti hai — jaise zero se divide karna — toh $a$ ke upar koi dot nahi hai, aur pen-stroke mein ek gap aa jaata hai yeh puchne se pehle ki approaching ke baare mein kya hai.

2. Arrow $x\to a$ aur "approaching" ka idea

Picture. Input axis par ek marker rakho aur use $a$ ki taraf crawl karne do, kabhi uspar landing kiye bina. Dekho ki marker ke crawl karne par dot ki height kya hoti hai. Arrow $\to$ woh crawling motion hai.

Topic ko yeh kyun chahiye. Continuity do alag cheezein compare karti hai: graph kahan heading hai versus kahan land karta hai. Arrow "heading" ko capture karta hai landing spot ko kabhi touch kiye bina — yahi woh trick hai jo $a$ par ek hole exist karne deti hai jabki graph phir bhi kisi sensible jagah head karta hai.

3. Limit ${\lim }_{x\to a}f(x)$

Picture. Jaise tumhara marker $a$ ki taraf slide karta hai, dot ki height ek horizontal level $L$ par press karti hai. Chahe $a$ par exactly ek hole ho (wahan koi dot nahi), woh height jis par woh press kar raha tha phir bhi $L$ hai. Limit woh "target height" padhta hai, point khud ko ignore karke.

Yeh tool kyun, sirf $f(a)$ kyun nahi? Kyunki $f(a)$ missing, galat, ya galat jagah ho sakta hai, jabki aas-paas ke graph ka trend abhi bhi clearly ek value ki taraf point karta hai. Limit woh tool hai jo trend ko single point se independently measure karta hai — exactly wahi jo humein holes aur jumps detect karne chahiye.

Is idea ki deeper build ke liye dekho Limits — definition and one-sided limits.

4. One-sided limits $L^{-}$ aur $L^{+}$

Picture. Do alag marchers vertical line $x=a$ ki taraf approach karte hain: ek left se, ek right se. Har ek woh height report karta hai jis par uski side press kar rahi hai. Agar dono same height report karte hain, toh two-sided limit exist karta hai aur woh height equal hai. Agar woh disagree karte hain, toh koi single two-sided limit nahi hai.

Topic ko yeh kyun chahiye. Jump discontinuity entirely $L^{-}\ne L^{+}$ se define hoti hai: dono marchers alag heights par land karte hain, ek stair-step. Approach ko do sides mein split kiye bina, tum literally ek jump describe nahi kar sakte. Dekho Piecewise functions jo yeh two-sided splits kahaan se aate hain.

5. Infinity $\pm \infty$ aur blow up

Picture. $x=a$ ke paas graph ek vertical line ko hug karta hai aur uske saath upar (ya neeche) rocket karta hai. Woh vertical line jise graph kabhi cross nahi karta woh ek vertical asymptote hai.

Topic ko yeh kyun chahiye. Infinite discontinuity exactly yeh rocket hai. Aur note karo ki signs har side par differ kar sakti hain: $\frac{1}{x-3}$ ke liye, $3$ ke thoda neeche ek tiny negative denominator $-\infty$ deta hai, $3$ ke thoda upar ek tiny positive denominator $+\infty$ deta hai. Dono signs handle karna hi wajah hai ki parent har side separately check karta hai. Dekho Vertical asymptotes and rational functions.

6. $\frac{0}{0}$ symptom aur cancellation

Picture. Numerator aur denominator dono curves same input $a$ par zero cross karti hain. $a$ par khud fraction meaningless hai (koi dot nahi), lekin $a$ ke bilkul paas ratio ek clean finite number par settle ho sakta hai — woh settled number limit hai.

Topic ko yeh kyun chahiye. Removable discontinuities almost always $\frac{0}{0}$ ke roop mein show up hoti hain. Ek common $(x-a)$ factor ko factor aur cancel karna woh height reveal karta hai jis ki taraf graph ja raha tha. Exactly isliye parent ke Example 1 mein $\frac{x^2-1}{x-1}=x+1$ factor kiya jaata hai (for $x\ne 1$) limit $2$ expose karne ke liye.

7. Standard limit $\frac{\sin x}{x}\to 1$

Plain words. Jaise angle $x$ (radians mein) $0$ ki taraf shrink karta hai, $\sin x$ aur $x$ ka ratio $1$ ki taraf press karta hai — chahe $x=0$ par fraction warning-light $\frac{0}{0}$ ho.

Topic ko yeh kyun chahiye. Parent ke Example 4 mein iska use hota hai yeh kahne ke liye ki limit exist karta hai aur $1$ equals hai, isliye ek function jo $g(0)=5$ set karta hai usmein ek removable flaw hai (limit exist karta hai, value galat jagah hai). Poori geometric build Standard limit sin(x)/x = 1 par hai — yahan tumhe sirf value $1$ par trust karna hai.

Foundations topic ko kaise feed karte hain

Ise top-down padho: function aur approach-arrow limit build karte hain; limit do sides mein split hoti hai; sides ka agreement ya disagreement, plus value $f(a)$, continuity ya teen tarah ki failures decide karta hai.

Equipment checklist

Khud ko test karo — tum parent note ke liye ready ho sirf tab jab har reveal obvious ho.

Connections

• Continuity — definition, types of discontinuity (removable, jump, infinite) (yeh page ise feed karta hai)
• Limits — definition and one-sided limits ($\lim$, $L^{-}$, $L^{+}$ ki machinery)
• Standard limit sin(x)/x = 1 (§7 mein imported fact)
• Piecewise functions (two-sided splits aur jumps ka source)
• Vertical asymptotes and rational functions ($\pm \infty$ blow-ups ka source)