Continuity and Differentiability

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

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

Types of Limits:

• Left-hand limit: $\lim_{x \to a^{-}} f(x)$ approaching from left
• Right-hand limit: $\lim_{x \to a^{+}} f(x)$ approaching from right

• Two-sided limit exists if and only if both one-sided limits exist and are equal

$$\lim_{x \to a^{-}} f(x) = \lim_{x \to a^{+}} f(x) = \lim_{x \to a} f(x)$$

Important Limit Theorems: ⭐

• $\lim_{x \to 0} (sin x)/x = 1$
• $\lim_{x \to 0} (1 - cos x)/x = 0$
• $lim_{x \to ∞} (1 + 1/x)^x = e$

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

3. Continuity

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

1. $f(a)$ is defined
2. $\lim_{x \to a} f(x)$ exists
3. $\lim_{x \to a} f(x) = f(a)$

Types of Discontinuity:

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

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

3. Infinite Discontinuity: At least one one-sided limit is infinite Example: $f(x) = 1/x$ 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 $f$ is continuous on $[a,b]$ and k is between $f(a)$ and $f(b)$, then there exists c in $(a,b)$ such that $f(c) = k$.

4. Differentiability

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

$$f’(a) = \lim_{h→0} [f(a+h) - f(a)]/h \quad \text{exists}$$

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