Continuity & Differentiability — Class 12

Introduction

English: This chapter builds foundational concepts in calculus: limits, continuity, and differentiability. You will learn precise definitions, important theorems (Intermediate Value Theorem, Rolle's theorem, Mean Value Theorem), proofs, and methods to sketch behaviour near points. Diagrams (unit-circle, triangle method) are embedded as inline SVG placeholders so you can replace them with exact SVGs. Keywords included for SEO: Class 12 Maths, continuity notes, differentiability notes, police bharti notes, railway notes, ssc notes, NEET notes, rsetu.link.

Marathi / मराठी (संक्षेप): हा chapter calculus चा पाया मजबूत करतो: Limits, Continuity आणि Differentiability. सिद्धांत, प्रमेय, पुरावे व पद्धती सोप्या टप्प्यात दिल्या आहेत. SVG diagrams इथे बदलता येतील.

Prerequisites

• Basic algebra and trigonometry
• Limit concept and standard limits (sin x / x, (1+x)^{1/x})
• Continuity intuition from graphs

Key Definitions (with precision)

1. Limit: \(\lim_{x\to a} f(x) = L\) if for every \(\epsilon>0\) there exists \(\delta>0\) s.t. \(|f(x)-L|<\epsilon\) when \(0<|x-a|<\delta\).
2. Continuity at a point: f is continuous at \(a\) iff \(\lim_{x\to a} f(x)=f(a)\).
3. Differentiability at a point: f is differentiable at \(a\) iff the limit \(\lim_{h\to0} \frac{f(a+h)-f(a)}{h}\) exists (finite).

Important Theorems (statements + proofs)

1. Sequential criterion for limits

Statement: \(\lim_{x\to a} f(x)=L\) iff for every sequence \(x_n\to a\) with \(x_n\neq a\), we have \(f(x_n)\to L\).

Proof sketch: Use epsilon-delta to show forward direction; for converse, convert the contrapositive. (Full epsilon-delta details included in full PDF export).

2. Continuity implies limit equals function value

Direct from the definition. If differentiable at a point then continuous there (proof: derivative limit existence implies function limit equality).

3. Rolle's Theorem

Statement: If f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then there exists c in (a,b) with f'(c)=0.

Sketch: Use Fermat's theorem on maxima/minima of continuous functions on closed interval.

4. Mean Value Theorem (MVT)

Statement and proof via Rolle's theorem. Consequences: monotonicity and derivative sign relationships.

5. Intermediate Value Property for derivatives (Darboux's theorem) — statement only

Essential formulas & shortcuts

• Derivative rules: (fg)' = f'g + fg', (f/g)' = (f'g - fg')/g^2
• Chain rule, product rule, quotient rule
• Trigonometric derivatives: (sin x)' = cos x, (cos x)' = -sin x
• Standard limits: \(\lim_{x\to0} \frac{\sin x}{x}=1\), \(\lim_{x\to0} \frac{1-\cos x}{x^2}=1/2\)

Solved Examples (8) — step-by-step

1. Example 1: Show f(x)=x^2 is continuous at 2.

   Solution: Use limit laws: \(\lim_{x\to2} x^2 = 4 = f(2)\). Hence continuous.

2. Example 2: Prove f(x)=|x| is differentiable at x≠0, not differentiable at 0.

   Solution: For x>0 and x<0, derivative is ±1 by definition. At 0, left derivative = -1, right derivative = 1 → not equal → not differentiable.

3. Example 3: Compute \(\lim_{x\to0} \frac{\sin x - x}{x^3}\).

   Solution: Use Taylor expansion: sin x = x - x^3/6 + o(x^3). So limit = -1/6.

4. Example 4: If f(x)=x^3-3x+1, show there exists c in (0,2) with f'(c)=0.

   Solution: f(0)=1, f(2)=3 → not equal; but check f(0)=1 and f(1)=-1 so by Rolle/MVT on [0,1] since f continuous and differentiable and f(0)≠f(1) use MVT between points where values change sign to find c with f'(c)=0. Compute f'(x)=3x^2-3; solve 3x^2-3=0 → x=±1 → x=1 in (0,2).

5. Example 5: Show that if f is differentiable at a, then f is continuous at a.

   Solution: Use definition of derivative; rewrite f(a+h)=f(a)+h f'(a)+o(h) then take limit as h→0.

6. Example 6 (Trick): Evaluate \(\lim_{x\to0} \frac{\sqrt{1+x}-1}{x}\).

   Solution: Multiply numerator and denominator by conjugate: \(\frac{(\sqrt{1+x}-1)(\sqrt{1+x}+1)}{x(\sqrt{1+x}+1)}=\frac{x}{x(\sqrt{1+x}+1)}=\frac{1}{\sqrt{1+x}+1}\) → limit 1/2.

7. Example 7 (Differentiability piecewise): f(x)=\begin{cases} x^2 & x\le 1 \\ 2x-1 & x>1 \end{cases} Check differentiability at x=1.

   Solution: Continuity: left f(1)=1, right f(1)=1 → continuous. Left derivative 2x at 1 → 2; right derivative =2 → equal ⇒ differentiable at 1.

8. Example 8 (Using MVT): Show |sin x| ≤ |x| for all real x using MVT.

   Solution: For x>0 apply MVT to sin on [0,x]; there exists c with (sin x - 0)/x = cos c ⇒ |sin x| = |x cos c| ≤ |x|.

Practice Questions (12)

1. Prove \(\lim_{x\to2} \frac{x^2-4}{x-2}=4\).
2. Is f(x)=x\sin(1/x) continuous at 0? Differentiable at 0?
3. Show f(x)=x^2\sin(1/x) (with f(0)=0) is differentiable at 0.
4. Find where f(x)=|x^3-1| is differentiable.
5. Use MVT to show if f'(x)=0 for all x then f is constant.
6. Evaluate \(\lim_{x\to0} \frac{e^x-1-x}{x^2}\).
7. Prove continuity of polynomial functions.
8. Find derivative of f(x)=x^x.
9. Show function with jump discontinuity is not integrable in Riemann sense on interval containing jump (short explanation).
10. Find values of a,b so piecewise function is continuous everywhere: f(x)=x+ a for x<1, f(x)=b for x=1, f(x)=2x-1 for x>1.
11. Use the definition to find derivative of sqrt(x) at x=4.
12. Prove that differentiability implies continuity with epsilon-delta.

Summary Table (Quick Revision)

Answer Key — Practice Qs (brief)

1. 4 (factor & cancel).
2. Continuous at 0; not differentiable at 0 (limit of derivative fails).
3. Differentiable at 0 (use limit; term vanishes as x→0).
4. Differentiate where inner expression ≠1 or sign changes; cusp at roots where derivative may fail.
5. By MVT, for any x,y there exists c with f'(c)=(f(y)-f(x))/(y-x); if f'=0 then f(y)=f(x) so constant.
6. 1/2 (Taylor: e^x=1+x+x^2/2+...).
7. Polynomials are continuous everywhere (limits by algebra).
8. Take ln: y=x^x ⇒ ln y = x ln x ⇒ y'/y = ln x +1 ⇒ y'=x^x(ln x+1).
9. Riemann integral requires boundedness and small oscillation; jump creates non-negligible oscillation → not integrable (informal).
10. Set left and right limits equal at 1: 1+a = b = 2*1-1 ⇒ a=0, b=1.
11. f'(4)=1/(2*√4)=1/4 via definition or known rule.
12. See Example 5; epsilon-delta via derivative linearization.