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THE CS NOTES

Continuity and Differentiability

1. Functions: Foundation
Section titled “1. Functions: Foundation”

A function f: A → B is a rule that assigns each element in set A (domain) exactly one element in set B (codomain).

Key Concepts:

  • Domain: Set of all possible input values
  • Range: Set of actual output values
  • Codomain: Set of all possible output values
2. Limits: The Building Block
Section titled “2. Limits: The Building Block”

We say lim⁡x→a+f(x)=L\lim_{x \to a^{+}}f(x) = L if f(x)f(x) gets arbitrarily close to LL as xx approaches a (from both sides).

Types of Limits:

  • Left-hand limit: lim⁡x→a−f(x)\lim_{x \to a^{-}} f(x) approaching from left
  • Right-hand limit: lim⁡x→a+f(x)\lim_{x \to a^{+}} f(x) approaching from right
  • Two-sided limit exists if and only if both one-sided limits exist and are equal

lim⁡x→a−f(x)=lim⁡x→a+f(x)=lim⁡x→af(x)\lim_{x \to a^{-}} f(x) = \lim_{x \to a^{+}} f(x) = \lim_{x \to a} f(x)

Important Limit Theorems: ⭐

  • lim⁡x→0(sinx)/x=1\lim_{x \to 0} (sin x)/x = 1
  • lim⁡x→0(1−cosx)/x=0\lim_{x \to 0} (1 - cos x)/x = 0
  • limx→∞(1+1/x)x=elim_{x \to ∞} (1 + 1/x)^x = e

Indeterminate Forms: 0/00/0, ∞/∞∞/∞, 0×∞,0×∞, ∞-∞,, 0⁰,, 1^∞,, ∞⁰$

3. Continuity
Section titled “3. Continuity”

Definition: A function f(x) is continuous at x = a if:

  1. f(a)f(a) is defined
  2. lim⁡x→af(x)\lim_{x \to a} f(x) exists
  3. lim⁡x→af(x)=f(a)\lim_{x \to a} f(x) = f(a)

Types of Discontinuity:

  1. Removable Discontinuity: Limit exists but ≠ f(a)f(a) or f(a)f(a) undefined Example: f(x)=(x2−1)/(x−1)f(x) = (x²-1)/(x-1) at x = 1

  2. Jump Discontinuity: Left and right limits exist but are unequal Example: f(x)=∣x∣/xf(x) = |x|/x at x = 0

  3. Infinite Discontinuity: At least one one-sided limit is infinite Example: f(x)=1/xf(x) = 1/x at x = 0

Properties of Continuous Functions:

  • Sum, difference, product of continuous functions is continuous
  • Quotient is continuous where denominator ≠ 0
  • Composition of continuous functions is continuous
  • All polynomial, exponential, logarithmic, trigonometric functions are continuous in their domains

Intermediate Value Theorem (IVT):

  • If ff is continuous on [a,b][a,b] and k is between f(a)f(a) and f(b)f(b), then there exists c in (a,b)(a,b) such that f(c)=kf(c) = k.
4. Differentiability
Section titled “4. Differentiability”

A function f(x)f(x) is differentiable at x=ax = a if:

\text{exists}$$ This limit is called the **derivative** at x = a. **Alternative Definition**: $f'(a) = \lim_{x→a} [f(x) - f(a)]/(x-a)$ **Geometric Interpretation**: The derivative represents the slope of the tangent line to the curve at that point. **Relationship Between Continuity and Differentiability**: - **If f is differentiable at a, then f is continuous at a** (differentiability ⟹ continuity) - **The converse is NOT true**: Continuous functions may not be differentiable - Example: f(x) = |x| is continuous at x = 0 but not differentiable there **Non-differentiable Points**: 1. **Sharp corners/cusps**: f(x) = |x| at x = 0 2. **Vertical tangents**: f(x) = x^(1/3) at x = 0 3. **Discontinuities**: Any point of discontinuity ##### **5. Differentiation Rules** **Basic Rules**: - **Power rule:** - $d/dx(x^n) = nx^(n-1)$ - **Constant rule:** - $d/dx(c) = 0$ - **Sum rule:** - $d/dx[f(x) + g(x)] = f'(x) + g'(x)$ - **Product rule:** - $d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$ - **Quotient rule:** - $d/dx[f(x)/g(x)] = [f'(x)g(x) - f(x)g'(x)]/[g(x)]²$ - **Chain rule:** - $d/dx[f(g(x))] = f'(g(x))·g'(x)$ ##### **6. Critical Points and First Derivative Test** **Critical Point**: A point $x = c$ is critical if: - $f'(c) = 0$, OR - $f'(c)$ does not exist **First Derivative Test** (for identifying local extrema): ⭐ At a critical point $x = c$ : - If $f'(x)$ changes from positive to negative at c, then f has a **local maximum** at c - If $f'(x)$ changes from negative to positive at c, then f has a **local minimum** at c - If $f'(x)$ does not change sign, then c is neither a local max nor min (could be an **inflection point**) **Sign Analysis**: - $f'(x) > 0$ on an interval ⟹ f is **increasing** on that interval - $f'(x) < 0$ on an interval ⟹ f is **decreasing** on that interval ##### **7. Local Maxima and Minima** **Local Maximum**: - f has a local maximum at $x = c$ if $f(c) ≥ f(x)$ for all x in some open interval containing c. - If $f'(c) = 0$ and $f''(c) < 0$ then f has a **local maximum** at c **Local Minimum**: - f has a local minimum at $x = c$ if $f(c) ≤ f(x)$ for all x in some open interval containing c. - If $f'(c) = 0$ and $f''(c) > 0$ then f has a **local maximum** at c **Global (Absolute) Maximum**: - $f(c) ≥ f(x)$ for all x in the domain **Global (Absolute) Minimum**: - $f(c) ≤ f(x)$ for all x in the domain **Note**: - **Fermat's Theorem**: If f has a local extremum at x = c and f'(c) exists, then f'(c) = 0. - Not every point where $f'(c) = 0$ is a local extremum (example: f(x) = x³ at x = 0) ##### **8. Second Derivative and Concavity** **Second Derivative**: $$f''(x) = d/dx[f'(x)]$$ **Concavity**: - $f''(x) > 0$ on an interval ⟹ f is **concave up** (curves upward like ∪) - $f''(x) < 0$ on an interval ⟹ f is **concave down** (curves downward like ∩) **Point of Inflection**: A point where the concavity changes (where the curve changes from concave up to concave down or vice versa). At inflection points, $f''(x) = 0$ or $f''(x) does not exist$. **Second Derivative Test** (for local extrema): At a critical point where f'(c) = 0: ⭐ - If $f''(c) > 0$, then f has a **local minimum** at c - If $f''(c) < 0$, then f has a **local maximum** at c - If $f''(c) = 0$, the test is **inconclusive** (use first derivative test) ##### **9. Mean Value Theorem (MVT)** **Rolle's Theorem**: - If f is continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a) = f(b)$, then there exists at least one c in $(a,b)$ such that $f'(c) = 0$. **Mean Value Theorem**: - If f is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists at least one c in $(a,b)$ such that: $f'(c) = [f(b) - f(a)]/(b-a)$ **Interpretation**: - There exists at least one point where the instantaneous rate of change equals the average rate of change. **Applications**: - Proving inequalities - Establishing relationships between functions and their derivatives - Showing that if f'(x) = 0 everywhere, then f is constant ##### **10. Monotonicity and Extrema** **Monotonic Functions**: - **Increasing**: - $f'(x) ≥ 0$ for all x in the interval (strictly increasing if $f'(x) > 0$) - **Decreasing**: - $f'(x) ≤ 0$ for all x in the interval (strictly decreasing if $f'(x) < 0$) **Finding Absolute Extrema on $[a,b]$**: 1. Find all critical points in $(a,b)$ 2. Evaluate $f$ at all critical points 3. Evaluate $f$ at endpoints $a$ and $b$ 4. The largest value is the absolute maximum, smallest is the absolute minimum **Extreme Value Theorem**: - If f is continuous on a closed interval $[a,b]$, then f attains both an absolute maximum and an absolute minimum on that interval. ##### **11. L'Hôpital's Rule** ⭐ For indeterminate forms 0/0 or ∞/∞: if $lim_{x→a} f(x)/g(x)$ gives $0/0$ or $∞/∞$, then: $$lim_{x→a} f(x)/g(x) = lim_{x→a} f'(x)/g'(x)$$ (provided the limit on the right exists) Can be applied repeatedly if needed. ##### **12. Summary of Relationships** **Sequential Flow**: 1. **Function** → 2. **Limit** → 3. **Continuity** → 4. **Differentiability** → 5. **Critical Points** → 6. **Extrema** **Key Hierarchy**: - Differentiable ⟹ Continuous ⟹ Limit exists - Continuous ⏸ Differentiable (not always) - f'(c) = 0 ⏸ Local extremum at c (not always) **Analysis Process for Finding Extrema**: 1. Find the domain 2. Find f'(x) 3. Find critical points (f'(x) = 0 or undefined) 4. Apply first derivative test or second derivative test 5. Check endpoints if on a closed interval 6. Identify local and global extrema