Continuity and Differentiability

Continuity and differentiability are the gatekeepers of calculus. A function is continuous if its graph has no breaks, jumps, or holes; a function is differentiable if we can draw a tangent line at every point—meaning the derivative exists everywhere. This chapter formalizes these intuitive ideas, introduces new classes of powerful functions (exponentials and logarithms), and explores the intricate relationship between continuity and differentiability. Understanding these concepts is essential for the applications of calculus in chapter-06-application-of-derivatives, chapter-07-integrals, and beyond.

Understanding Continuity

Intuitive Definition: A function f is continuous at point x = a if the graph has no break, jump, or hole at a—you could draw it without lifting your pencil.

Formal Definition: f is continuous at x = a if:

1. f(a) is defined
2. lim(x→a) f(x) exists
3. lim(x→a) f(x) = f(a)

In other words, the limit as you approach a equals the function value at a.

Left and right continuity: If lim(x→a⁻) f(x) = f(a), f is left-continuous. If lim(x→a⁺) f(x) = f(a), f is right-continuous. A function is continuous at a if it's both left and right continuous.

Types of Discontinuities

• Removable discontinuity: A hole in the graph; the limit exists but f(a) is undefined or doesn't equal the limit. Example: f(x) = (x² - 1)/(x - 1) at x = 1 (can be "fixed" by redefining f(1) = 2)
• Jump discontinuity: The left and right limits exist but differ. Example: the floor function at integers
• Infinite discontinuity: The function approaches ±∞ as x approaches a. Example: f(x) = 1/x at x = 0

Building from Class 11

In Class 11, you computed derivatives of polynomial and trigonometric functions using the limit definition. This chapter reveals that differentiability requires continuity first: if a function is differentiable at a point, it must be continuous there. The converse is false—continuous functions need not be differentiable (sharp corners, cusps).

Introducing New Functions: Exponential and Logarithmic

Exponential Functions

f(x) = aˣ (where a > 0, a ≠ 1) grows exponentially. The natural exponential f(x) = eˣ is special: its derivative equals itself!

• Derivative: d/dx[eˣ] = eˣ
• Continuity: eˣ is continuous everywhere
• Properties: e⁰ = 1, eˣ → ∞ as x → ∞, eˣ → 0 as x → -∞
• Inverse: The natural logarithm ln(x) is the inverse of eˣ

Logarithmic Functions

f(x) = log_a(x) (where a > 0, a ≠ 1) is the inverse of aˣ. The natural logarithm f(x) = ln(x) is particularly important:

• Derivative: d/dx[ln(x)] = 1/x
• Continuity: ln(x) is continuous for x > 0
• Domain: Only defined for x > 0
• Properties: ln(1) = 0, ln(xy) = ln(x) + ln(y), ln(eˣ) = x

Relationship Between e and ln

The exponential and logarithmic functions are inverses:

• e^(ln(x)) = x for all x > 0
• ln(eˣ) = x for all x ∈ ℝ

Understanding Differentiability

Definition: f is differentiable at x = a if the derivative f'(a) = lim(h→0) [f(a+h) - f(a)]/h exists (as a finite number).

Geometric meaning: The function has a well-defined tangent line with slope f'(a) at point (a, f(a)). There are no sharp corners or cusps.

Theorem: If f is differentiable at a, then f is continuous at a. (Differentiability implies continuity, but not vice versa.)

Example of Continuity Without Differentiability

f(x) = |x| is continuous at x = 0 but not differentiable there:

• Left derivative: lim(h→0⁻) |h|/h = -1
• Right derivative: lim(h→0⁺) |h|/h = 1
• They don't match, so f'(0) doesn't exist

Derivatives of New Functions

Using the chain rule and the derivatives of exponential/logarithmic functions:

• d/dx[aˣ] = aˣ · ln(a)
• d/dx[log_a(x)] = 1/(x · ln(a))
• d/dx[ln(f(x))] = f'(x)/f(x) (logarithmic differentiation)

Logarithmic differentiation is powerful for products and quotients: If y = (x+1)²(x-2)³, take ln(y) = 2ln(x+1) + 3ln(x-2), then differentiate to find y'.

Connections to Other Topics

• chapter-01-relations-and-functions: Functions must be continuous to have well-behaved inverses
• chapter-02-inverse-trigonometric-functions: Continuity and differentiability of inverse trig functions
• chapter-06-application-of-derivatives: Uses differentiability to find maxima, minima, and rates of change
• chapter-07-integrals: Continuous functions on closed intervals are integrable

Socratic Questions

1. Consider the function f(x) = |x - 1|. Is it continuous at x = 1? Is it differentiable at x = 1? Why or why not? What does the graph look like at that point?

1. Explain why every differentiable function must be continuous, but not every continuous function is differentiable. Can you sketch a function that is continuous but has a point where it's not differentiable?

1. The derivative of eˣ is eˣ itself. Why is this property unique and so important? How does this compare to the derivatives of other exponential functions like 2ˣ?

1. In logarithmic differentiation, we take ln(y) instead of differentiating y directly. Why is this technique useful for products and quotients? What advantage does it provide?

1. If a function f is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), what does the Mean Value Theorem guarantee about the derivative somewhere in (a, b)? Why is this powerful?