Continuity and Differentiability
1. Continuity - No Break in Graph
Continuity at x=c means you can draw graph through c without lifting pen. Three conditions: f(c) defined, limit exists (LHL=RHL), limit = f(c). If any fails, discontinuous. Example f(x)=|x| continuous at 0 because limit 0 = f(0)=0. f(x)=1/x discontinuous at 0 not defined. f(x)=[x] greatest integer discontinuous at integers jump. Check continuity: find LHL lim x→c- f(x), RHL lim x→c+ f(x), compare with f(c).
Continuity at c: f(c) defined, lim x→c f(x) exists, lim = f(c). LHL=RHL=f(c). Check. |x| continuous at 0, 1/x not continuous at 0, [x] discontinuous at integers.
• Definition: LHL = RHL = f(c) at x=c
• f(c) must be defined, limit must exist
• Example: |x| continuous at 0: limit 0 = f(0)
• Discontinuous: 1/x at 0 not defined, [x] at integers jump
• Steps: Find LHL, RHL, f(c), compare
2. Types of Discontinuity - Why Break Happens
Removable discontinuity: limit exists but f(c) different or undefined, hole can be filled. Example f(x)=(x²-1)/(x-1) limit 2 at x=1 but not defined at 1. Jump discontinuity: LHL≠RHL, graph jumps, like [x]. Infinite discontinuity: limit ±∞ vertical asymptote like 1/x at 0. Oscillating: sin(1/x) near 0 oscillates limit not exist. Graph tells type: hole, jump, asymptote, wiggly.
Types: Continuous no break, Removable hole limit exists f(c) different, Jump LHL≠RHL step, Infinite limit ±∞ asymptote, Oscillating no limit oscillates. Graph behaviours.
• Continuous: No break limit=f(c)
• Removable: Limit exists but f(c) undefined/different hole
• Jump: LHL≠RHL step e.g. [x]
• Infinite: Limit ±∞ asymptote e.g. 1/x at 0
• Oscillating: sin(1/x) at 0 no limit
• Visual: hole, jump, asymptote, wiggle
3. Differentiability - Smoothness
Differentiability at c means derivative exists, graph has unique tangent no sharp corner. Definition: Left derivative = Right derivative. Left derivative lim h→0- [f(c+h)-f(c)]/h, Right derivative lim h→0+ same. If LHD=RHD finite, differentiable. Example |x| at 0: LHD=-1 RHD=+1 not equal so not differentiable at 0 (sharp corner). x² differentiable everywhere. Differentiable ⇒ Continuous but Continuous ⇏ Differentiable. So continuity necessary but not sufficient.
Differentiable at c if LHD=RHD finite. LHD=lim h→0- [f(c+h)-f(c)]/h, RHD=lim h→0+ same. |x| at 0 LHD=-1 RHD=1 not differentiable sharp corner. Differentiable ⇒ Continuous, converse false.
• Definition: LHD=RHD finite at c
• LHD=lim h→0- [f(c+h)-f(c)]/h, RHD=lim h→0+
• Example |x| at 0 LHD=-1 RHD=1 not differentiable corner
• x² differentiable everywhere smooth
• Theorem: Differentiable ⇒ Continuous, but Continuous ⇏ Differentiable
4. Exponential and Logarithmic Derivatives
Derivatives of e^x is e^x itself, exponential never changes. Derivative of ln|x| is 1/x. Derivative of a^x = a^x ln a. Derivative of log_a x = 1/(x ln a). Derivative of e^{f(x)} = e^{f(x)} f'(x) chain rule. Example d/dx e^{x²}=e^{x²}·2x. d/dx ln(x²+1)=2x/(x²+1). These are most used in log differentiation. Remember e^x derivative same, ln x derivative 1/x.
d/dx e^x=e^x, d/dx ln|x|=1/x, d/dx a^x=a^x ln a, d/dx log_a x=1/(x ln a). Chain: d/dx e^{f(x)}=e^{f(x)}f'(x), d/dx ln f(x)=f'(x)/f(x).
• d/dx e^x=e^x, d/dx ln|x|=1/x
• d/dx a^x=a^x ln a, d/dx log_a x=1/(x ln a)
• Chain: e^{f(x)} → e^{f(x)}·f'(x)
• ln f(x) → f'(x)/f(x)
• Example: e^{x²} →2x e^{x²}, ln(x²+1)→2x/(x²+1)
• e^x never changes
5. Logarithmic Differentiation - For x^x Type
When function is like y=x^x or y=(sin x)^{cos x} or product/quotient with many factors, take log both sides: ln y = ln(...). Then differentiate implicitly: (1/y) y' = derivative of ln RHS. Solve y' = y × ... Example y=x^x ln y=x ln x differentiate (1/y) y' = ln x + x·1/x = ln x+1 so y'=x^x(ln x+1). For product y=(x)(x+1)(x+2) log makes sum easy.
For y=[f(x)]^{g(x)} or huge product: ln y = g(x) ln f(x), differentiate: y'/y = g'(x)ln f(x)+g(x)f'(x)/f(x), y'=y[ ]. For product log converts to sum.
• Use for y=x^x, y=(sin x)^{cos x}, huge product/quotient
• Step1: ln y = g(x) ln f(x)
• Step2: Differentiate (1/y) y' = g' ln f + g f'/f
• Step3: y'=y[ ]
• Example x^x → ln y=x ln x → y'/y=ln x+1 → y'=x^x(ln x+1)
• Product (x)(x+1)... log→sum
6. Parametric Functions - x and y both in t
Parametric: x=f(t), y=g(t) where t parameter, like x= a cos t, y= a sin t circle. Then dy/dx = (dy/dt)/(dx/dt) provided dx/dt≠0. Example x= t², y= t³, dx/dt=2t, dy/dt=3t² so dy/dx=3t²/2t=3t/2. Second order also parametric. Used for projectile motion x=v0 cosθ t, y=... Find slope at any t. Chain rule essence.
Parametric x=f(t), y=g(t) dy/dx=(dy/dt)/(dx/dt) dx/dt≠0. Example x=t² y=t³ dy/dx=3t/2. For second derivative d²y/dx² = d/dt(dy/dx) / (dx/dt).
• x=f(t), y=g(t), t parameter
• dy/dx=(dy/dt)/(dx/dt) provided dx/dt≠0
• Example x=t² y=t³ → dx/dt=2t dy/dt=3t² → dy/dx=3t/2
• Circle: x=a cos t y=a sin t dy/dx=-cot t
• Second order: d²y/dx² = [d/dt(dy/dx)]/(dx/dt)
• Use: Projectile, curve slope
7. Second Order Derivative - Derivative of Derivative
Second order derivative d²y/dx² is derivative of first derivative dy/dx, tells rate of change of slope, concavity, acceleration. Notation y'' or f''. Example y=x³ dy/dx=3x² d²y/dx²=6x. If d²y/dx²>0 concave up cup shape slope increasing, if <0 concave down cap shape slope decreasing, if =0 possible inflection. In physics d²x/dt²=acceleration. Example y=x² d²y/dx²=2 always concave up.
d²y/dx² = d/dx(dy/dx). Measures concavity, slope change, acceleration. >0 concave up slope ↑, <0 concave down slope ↓, =0 possible inflection. Example x² d²y/dx²=2>0 concave up.
• Definition: d²y/dx² = d/dx(dy/dx) derivative of derivative
• Meaning: Rate of change of slope, concavity
• >0 concave up cup slope increasing
• <0 concave down cap slope decreasing
• =0 possible inflection point
• Physics: d²x/dt²=acceleration, Example x³ →6x
8. Real Life Usage - Where Continuity Differentiability Lives
Continuity ensures no sudden jump, used in engineering designs bridge curve must be continuous. Differentiability ensures smoothness no sharp corner, roads curves need differentiable for smooth driving. Exponential growth bacteria population e^{kt} derivative gives growth rate. Logarithmic differentiation used in complex formulas. Second derivative used in acceleration, concavity of profit graphs, bending of beams. Parametric used in animation, rocket trajectory x(t), y(t).
Real: Bridge continuous, road smooth differentiable, exponential population growth, log diff for complex, second derivative acceleration and profit concavity, parametric animation rocket trajectory.
• Continuity: Bridge curve no break, no jump in design
• Differentiability: Road smooth no sharp corner driving
• Exponential: Bacteria e^{kt} growth rate
• Log diff: Complex product formulas
• Second derivative: Acceleration, profit concavity, beam bending
• Parametric: Animation, rocket x(t) y(t) trajectory
• Continuity at c: LHL=RHL=f(c), f(c) defined limit exists equal
• Types: Continuous no break, Removable hole, Jump LHL≠RHL, Infinite ±∞ asymptote, Oscillating no limit
• Differentiability: LHD=RHD finite, LHD=lim h→0- [f(c+h)-f(c)]/h, RHD=lim h→0+
• |x| at 0: Continuous but LHD=-1 RHD=1 not differentiable corner
• Theorem: Differentiable ⇒ Continuous, converse false
• Exp Log: e^x→e^x, ln|x|→1/x, a^x→a^x ln a, log_a x→1/(x ln a), e^{f}→e^{f} f', ln f→f'/f
• Log Diff: y=f^g ln y=g ln f → y'/y=g' ln f+g f'/f → y'=y[ ] for x^x type
• Parametric: x=f(t) y=g(t) dy/dx=(dy/dt)/(dx/dt), d²y/dx²=[d/dt(dy/dx)]/(dx/dt)
• Second Order: d²y/dx²=d/dx(dy/dx), >0 concave up, <0 concave down, =0 inflection, acceleration
• Real: Bridge continuous, road differentiable smooth, bacteria growth, profit concavity, rocket trajectory