Understanding Continuous Functions

Continuity on an Open Interval

• A function 𝑓 is continuous on an open interval if it is continuous at every point in that interval.
• Polynomial functions are continuous on any open interval.

Right-Hand Continuity

• A function 𝑓 is continuous from the right at 𝑏 if:
  • 𝑓(𝑏) exists.
  • lim_(𝑥→𝑏+) 𝑓(𝑥) exists.
  • lim_(𝑥→𝑏+) 𝑓(𝑥) = 𝑓(𝑏).

Left-Hand Continuity

• A function 𝑓 is continuous from the left at 𝑏 if:
  • 𝑓(𝑏) exists.
  • lim_(𝑥→𝑏−) 𝑓(𝑥) exists.
  • lim_(𝑥→𝑏−) 𝑓(𝑥) = 𝑓(𝑏).

Continuity on a Closed Interval

• A function 𝑓 is continuous on a closed interval [𝑎, 𝑏] if:
  • 𝑓 is continuous on the open interval (𝑎, 𝑏).
  • 𝑓 is continuous from the right at 𝑎.
  • 𝑓 is continuous from the left at 𝑏.
• Polynomials are continuous on any closed interval.

Continuity on a Half-Open Interval

• A function 𝑓 with a domain including [𝑎, 𝑏) is continuous on [𝑎, 𝑏) if:
  • 𝑓 is continuous on the open interval (𝑎, 𝑏).
  • 𝑓 is continuous from the right at 𝑎.
• A function 𝑓 with a domain including (𝑎, 𝑏] is continuous on (𝑎, 𝑏] if:
  • 𝑓 is continuous on the open interval (𝑎, 𝑏).
  • 𝑓 is continuous from the left at 𝑏.
• Polynomial functions are continuous on any half-open interval.

Intermediate Value Theorem (IVT)

• If 𝑓 is continuous on [𝑎, 𝑏] and 𝑓(𝑎) ≠ 𝑓(𝑏), then for any 𝑧 between 𝑓(𝑎) and 𝑓(𝑏), there exists 𝑐 between 𝑎 and 𝑏 such that 𝑓(𝑐) = 𝑧.

Root Existence Example

• To prove that ℎ(𝑥) = 𝑥² – 𝑥 – 6 has a root on [–4,0]:
  • ℎ is a polynomial, so continuous on [−4,0].
  • ℎ(−4) = 14 and ℎ(0) = −6.
  • Because ℎ(0) < 0 < ℎ(−4), there exists 𝑐 ∈ (−4,0) such that ℎ(𝑐) = 0 by the IVT.

Squeeze Theorem

• If 𝑓(𝑥) ≤ 𝑔(𝑥) ≤ ℎ(𝑥) for all 𝑥 in an open interval 𝐾 (except possibly at 𝑎), and lim_(𝑥→𝑎) 𝑓(𝑥) and lim_(𝑥→𝑎) ℎ(𝑥) exist and equal 𝐿, then lim_(𝑥→𝑎) 𝑔(𝑥) also exists and equals 𝐿.

Special Trigonometric Limit

• lim_(𝜃→0) (sin 𝜃)/𝜃 = 1

Continuity of Sine and Cosine

• The sine and cosine functions are continuous at 0 and every real number.

Limit of (1 - cos θ)/θ

• lim_(𝜃→0) (1 − cos 𝜃)/𝜃 = 0

Equivalent Limit Statements

• lim_(𝑥→𝑎) 𝑓(𝑥) = 𝐿 is equivalent to:
  • lim_(𝑥→0) 𝑓(𝑥 + 𝑎) = 𝐿
  • lim_(𝑥→𝑎) [𝑓(𝑥) − 𝐿] = 0

Continuity of Trigonometric Functions

• Tangent, cotangent, secant, and cosecant are continuous on their domains.

Secant Line

• A line passing through two points on a curve.

Tangent Line Definition

• If 𝑓 is continuous at 𝑥₁, the tangent line to the graph of 𝑓 at 𝑃(𝑥₁, 𝑓(𝑥₁)) is:
  • The line through 𝑃 with slope 𝑚(𝑥₁) = lim_(Δ𝑥→0) (𝑓(𝑥₁ + Δ𝑥) − 𝑓(𝑥₁))/Δ𝑥, if this limit exists.
  • The line 𝑥 = 𝑥₁ if lim_(Δ𝑥→0+) (𝑓(𝑥₁ + Δ𝑥) − 𝑓(𝑥₁))/Δ𝑥 is +∞ or −∞, and lim_(Δ𝑥→0-) (𝑓(𝑥₁ + Δ𝑥) − 𝑓(𝑥₁))/Δ𝑥 is +∞ or −∞.
• The point 𝑃(𝑥₁, 𝑓(𝑥₁)) is the point of tangency.

Normal Line

• A line perpendicular to the tangent line of 𝑓 at point 𝑃.

Slope of a Graph

• The slope of the graph of 𝑓 at a point 𝑃 is the slope of the tangent line at 𝑃.

Derivative Definition

• The derivative of 𝑓, denoted by 𝑓', is 𝑓'(𝑥) = lim_(Δ𝑥→0) (𝑓(𝑥 + Δ𝑥) − 𝑓(𝑥))/Δ𝑥, if the limit exists.

Derivative at a Point

• If 𝑥₁ is in the domain of 𝑓, then 𝑓'(𝑥₁) = lim_(Δ𝑥→0) (𝑓(𝑥₁ + Δ𝑥) − 𝑓(𝑥₁))/Δ𝑥, if the limit exists, and equals 𝑚(𝑥₁).
• The derivative of 𝑓 evaluated at 𝑥₁ is the slope of the tangent line at (𝑥₁, 𝑓(𝑥₁)).

Derivative Notations

• Leibniz: 𝑑𝑦/𝑑𝑥 = 𝑑/𝑑𝑥(𝑓(𝑥))
• Lagrange: 𝑓'(𝑥)
• Euler: 𝐷𝑓

Derivative Example

• Given 𝑔(𝑥) = 𝑥², then 𝑔'(𝑥) = 2𝑥.

Tangent Line Equation Example

• For 𝑔(𝑥) = 𝑥² at the point (1, 𝑔(1)), the tangent line equation is 𝑦 − 1 = 2(𝑥 − 1).

Limit-Based Derivative Caveat

• The derivative, being a limit, may not exist depending on the value of x.

Derivative of 1/x Example

• If ℎ(𝑥) = 1/𝑥, then ℎ'(𝑥) = −1/(𝑥(𝑥 + Δ𝑥)), which is undefined at 𝑥 = 0.

Alternative Derivative Formula

• By letting 𝑥₁ + Δ𝑥 = 𝑥, the derivative can be defined as 𝑓'(𝑥₁) = lim_(𝑥→𝑥₁) (𝑓(𝑥) − 𝑓(𝑥₁))/(𝑥 − 𝑥₁).

Differentiability

• A function 𝑓 is differentiable at 𝑥₀ if 𝑓'(𝑥₀) exists.
• If 𝑓 is differentiable at every point on an open interval (𝑎, 𝑏), then 𝑓 is differentiable on (𝑎, 𝑏).
• If 𝑓 is differentiable at every point on its domain, then 𝑓 is a differentiable function.

Differentiability Implies Continuity

• If 𝑓 is differentiable at 𝑥₁, then 𝑓 is continuous at 𝑥₁.
• The converse is not necessarily true.

One-Sided Derivatives

• The derivative from the right of 𝑓 at 𝑥₁ is 𝑓+'(𝑥₁) = lim_(𝑥→𝑥₁+) (𝑓(𝑥) − 𝑓(𝑥₁))/(𝑥 − 𝑥₁).
• The derivative from the left of 𝑓 at 𝑥₁ is 𝑓−'(𝑥₁) = lim_(𝑥→𝑥₁-) (𝑓(𝑥) − 𝑓(𝑥₁))/(𝑥 − 𝑥₁).

Non-Differentiability

• A function 𝑓 can fail to be differentiable at 𝑎 if:
  • 𝑓 is discontinuous at 𝑥 = 𝑎.
  • 𝑓 has a vertical tangent line at 𝑥 = 𝑎.
  • 𝑓 does not have a tangent line at 𝑥 = 𝑎.

Basic Derivative Rules

• If 𝑓(𝑥) = 𝑐 (constant), then 𝑓'(𝑥) = 0.
• If 𝑓(𝑥) = 𝑥ⁿ, then 𝑓'(𝑥) = 𝑛𝑥ⁿ⁻¹.
• If 𝑔(𝑥) = 𝑐𝑓(𝑥), then 𝑔'(𝑥) = 𝑐𝑓'(𝑥).

Sum Rule of Differentiation

• If ℎ(𝑥) = 𝑓(𝑥) + 𝑔(𝑥), then ℎ'(𝑥) = 𝑓'(𝑥) + 𝑔'(𝑥) for differentiable functions 𝑓 and 𝑔.

Product Rule of Differentiation

• If ℎ(𝑥) = 𝑓(𝑥)𝑔(𝑥), then ℎ'(𝑥) = 𝑓(𝑥)𝑔'(𝑥) + 𝑔(𝑥)𝑓'(𝑥) for differentiable functions 𝑓 and 𝑔.

Quotient Rule of Differentiation

• If ℎ(𝑥) = 𝑓(𝑥)/𝑔(𝑥) and 𝑔(𝑥) ≠ 0, then ℎ'(𝑥) = (𝑔(𝑥)𝑓'(𝑥) − 𝑓(𝑥)𝑔'(𝑥))/[𝑔(𝑥)]².

Power Rule for Negative Exponents

- If 𝑓(𝑥) = 𝑥⁻ⁿ, where −𝑛 is a negative integer and 𝑥 ≠ 0, then 𝑓'(𝑥) = −𝑛𝑥⁻ⁿ⁻¹.

Chain Rule of Differentiation

- If 𝑔 is differentiable at 𝑥 and 𝑓 is differentiable at 𝑔(𝑥), then (𝑓 ∘ 𝑔)'(𝑥) = 𝑓'(𝑔(𝑥)) ⋅ 𝑔'(𝑥).

General Power Rule

- If 𝑓(𝑥) = 𝑥^(𝑚/𝑛), where 𝑚 and 𝑛 are integers and 𝑛 ≠ 0, then 𝑓'(𝑥) = (𝑚/𝑛) 𝑥^((𝑚/𝑛) − 1).

Derivatives of Trigonometric Functions

- d/dx(sin 𝑢) = cos 𝑢 ⋅ du/dx
- d/dx(cos 𝑢) = −sin 𝑢 ⋅ du/dx
- d/dx(tan 𝑢) = sec² 𝑢 ⋅ du/dx
- d/dx(cot 𝑢) = −csc² 𝑢 ⋅ du/dx
- d/dx(sec 𝑢) = sec 𝑢 tan 𝑢 ⋅ du/dx
- d/dx(csc 𝑢) = −csc 𝑢 cot 𝑢 ⋅ du/dx

Derivatives of Inverse Trigonometric Functions

• d/dx(sin⁻¹ 𝑢) = 1/(√(1 − 𝑢²)) ⋅ du/dx
• d/dx(cos⁻¹ 𝑢) = -1/(√(1 − 𝑢²)) ⋅ du/dx
• d/dx(tan⁻¹ 𝑢) = 1/(1 + 𝑢²) ⋅ du/dx
• d/dx(cot⁻¹ 𝑢) = -1/(1 + 𝑢²) ⋅ du/dx
• d/dx(sec⁻¹ 𝑢) = 1/(|𝑢|√(𝑢² − 1)) ⋅ du/dx
• d/dx(csc⁻¹ 𝑢) = -1/(|𝑢|√(𝑢² − 1)) ⋅ du/dx

Derivatives of Exponential and Logarithmic Functions

• d/dx(logₐ 𝑢) = 1/((ln 𝑎)𝑢) ⋅ du/dx
• d/dx(𝑎ᵘ) = (ln 𝑎)𝑎ᵘ ⋅ du/dx
• d/dx(𝑒ᵘ) = 𝑒ᵘ ⋅ du/dx

Higher Order Derivatives

• If a function 𝑓 is differentiable, its derivative 𝑓' is called the first derivative of f. The derivative of the first derivative is called the second derivative or f double prime and denoted as 𝑓''. The process can be repeated as derivatives are differntiable
• Leibniz notation: 𝑑²/(𝑑𝑥²) 𝑓(𝑥), 𝑑³/(𝑑𝑥³) 𝑓(𝑥), 𝑑⁴/(𝑑𝑥⁴) 𝑓(𝑥), ... , 𝑑ⁿ/(𝑑𝑥ⁿ) 𝑓(𝑥)
• Lagrange's notation: 𝑓''(𝑥), 𝑓'''(𝑥), 𝑓^(4)(𝑥), ..., 𝑓^(𝑛)(𝑥)
• Euler's notation: 𝐷²𝑓, 𝐷³𝑓, 𝐷⁴𝑓, ..., 𝐷ⁿ𝑓.

Rolle's Theorem

• If a function 𝑓 satisfies the following conditions: is continuous on a closed interval [𝑎, 𝑏], is differentiable with an open interval (𝑎, 𝑏), and 𝑓(𝑎) = 0 = 𝑓(𝑏); then a number 𝑐 ∈ (𝑎, 𝑏) exists where 𝑓'(𝑐) = 0.

Mean Value Theorem (MVT)

• For a function 𝑓 continuous on a closed interval [𝑎, 𝑏] and differentiable on the open interval (𝑎, 𝑏), there exists a number 𝑐 ∈ (𝑎, 𝑏) such that 𝑓'(𝑐) = (𝑓(𝑏) − 𝑓(𝑎))/(𝑏 − 𝑎).