Unit 2: Functions, limit and continuity

Functions describe how one quantity depends on another; limits capture the value a function approaches near a point; continuity records where a function has no breaks. This unit builds from the object (function) to its local behaviour (limit) to a smoothness property defined by limits (continuity).

• Set-theoretic base: A function pairs each input with exactly one output; limits and continuity are then defined on functions from subsets of ℝ to ℝ.
• Domain and codomain convention: Unless stated, domain is the largest subset of ℝ on which the rule makes sense (real functions).
• Local versus global: Limits and continuity are local (about a point); algebra rules let us build complex cases from simple ones.
• Standard notation: f: A → B, y = f(x), lim_{x→a} f(x) = L, f continuous at a.

II. Functions and Their Structure

Orientation: a function is a deterministic rule of correspondence, and this section covers what functions are, their common families, and how they combine arithmetically.

A. functions

A function f from set A to set B assigns to every element of A a unique element of B.

• Formal statement: f: A → B where for each x ∈ A there is exactly one y = f(x) ∈ B.
• Domain: the set A of all admissible inputs; e.g. for f(x)=1/x, domain is ℝ \ {0}.
• Codomain and range: B is the codomain; range is {f(x) : x ∈ A}, the actual outputs; for f(x)=x², range is [0,∞).
• Vertical line test: a graph is a function iff no vertical line meets it twice.
• Image example: for f(x)=2x+1 on A={0,1,2}, image is {1,3,5}.

B. some specific types of functions

Certain functions recur so often they carry standard forms and graphs.

• Identity function: f(x)=x; graph is the line y=x.
• Constant function: f(x)=c; graph is a horizontal line.
• Polynomial function: f(x)=aₙxⁿ + … + a₁x + a₀; e.g. f(x)=x²−3x+2.
• Rational function: f(x)=p(x)/q(x) with q(x)≠0; e.g. (x+1)/(x−2), domain excludes x=2.
• Modulus function: f(x)=|x|, equal to x for x≥0 and −x for x<0.
• Signum function: sgn(x) = 1, 0, −1 for x>0, x=0, x<0.
• Greatest integer function: f(x)=[x], the largest integer ≤ x; [2.7]=2, [−1.3]=−2.
• Exponential and logarithmic: f(x)=aˣ (a>0) and f(x)=logₐx (x>0), inverse to each other.
• Trigonometric: sin x, cos x (period 2π), tan x (period π).

C. algebra of real functions

Real functions with common domain combine like numbers, point by point.

• Sum/difference: (f ± g)(x) = f(x) ± g(x) on D_f ∩ D_g.
• Scalar multiple: (αf)(x) = α·f(x) for real α.
• Product: (fg)(x) = f(x)·g(x).
• Quotient: (f/g)(x) = f(x)/g(x) where g(x) ≠ 0.
• Worked example: with f(x)=x², g(x)=x+1:
  
  TEXT

    (f+g)(x) = x² + x + 1
    (fg)(x)  = x²(x+1) = x³ + x²
    (f/g)(x) = x²/(x+1),  x ≠ −1

III. Limits

Orientation: the limit of f at a is the single value f(x) approaches as x approaches a, whether or not f(a) is defined. This section states the concept, its algebra, and its evaluation for the main function families.

A. limits

The limit describes tendency near a point, ignoring the point itself.

• Definition (intuitive): lim_{x→a} f(x) = L if f(x) gets arbitrarily close to L as x nears a, x ≠ a.
• One-sided limits: left limit lim_{x→a⁻} and right limit lim_{x→a⁺}.
• Existence criterion: the limit exists iff lim_{x→a⁻} f(x) = lim_{x→a⁺} f(x).
• Example of failure: for f(x)=|x|/x, left limit = −1, right limit = +1, so lim_{x→0} does not exist.
• Independence from f(a): lim_{x→2}(x²−4)/(x−2) = 4 though f(2) is undefined.

B. algebra of limits

If lim_{x→a} f(x) and lim_{x→a} g(x) both exist, limits distribute over arithmetic.

• Sum: lim (f+g) = lim f + lim g.
• Difference: lim (f−g) = lim f − lim g.
• Product: lim (fg) = (lim f)(lim g).
• Quotient: lim (f/g) = (lim f)/(lim g), provided lim g ≠ 0.
• Scalar: lim (α f) = α · lim f.
• Constant: lim_{x→a} c = c; basic: lim_{x→a} x = a.
• Example: lim_{x→3}(2x+5) = 2(3)+5 = 11.

C. limits of polynomials and rational functions

Polynomials are evaluated by substitution; rational functions need care when the denominator vanishes.

• Polynomial rule: if p(x) is a polynomial, lim_{x→a} p(x) = p(a) (direct substitution).
  • Example: lim_{x→2}(x²−3x+1) = 4−6+1 = −1.
• Rational, denominator nonzero: lim_{x→a} p(x)/q(x) = p(a)/q(a) when q(a) ≠ 0.
• Indeterminate 0/0: factor and cancel the common term, then substitute.
  
  TEXT

    lim_{x→2} (x²−4)/(x−2)
        = lim_{x→2} (x−2)(x+2)/(x−2)
        = lim_{x→2} (x+2) = 4

• Limit at infinity: divide by the highest power; lim_{x→∞}(3x²+1)/(x²−x) = 3.

D. limits of trigonometric functions

Trigonometric limits rest on continuity of sine and cosine plus one standard limit.

• Direct substitution (where continuous): lim_{x→a} sin x = sin a, lim_{x→a} cos x = cos a.
• Fundamental limit:
  
  TEXT

    lim_{x→0} (sin x)/x = 1        (x in radians)

• Companion results:
  • lim_{x→0} (tan x)/x = 1.
  • lim_{x→0} (1 − cos x)/x = 0.
  • lim_{x→0} (1 − cos x)/x² = 1/2.
• Worked example:
  
  TEXT

    lim_{x→0} (sin 3x)/x = lim_{x→0} 3·(sin 3x)/(3x) = 3·1 = 3

IV. Continuity

Orientation: a function is continuous where its graph can be drawn without lifting the pen; formally, the limit exists and equals the function value. This section defines continuity and shows how continuous functions combine.

A. continuity

Continuity at a point links the local limit to the actual value there.

• Definition: f is continuous at a if all three hold:
  
  TEXT

    (i)  f(a) is defined
    (ii) lim_{x→a} f(x) exists
    (iii) lim_{x→a} f(x) = f(a)

• Continuity on an interval: f is continuous on (c,d) if continuous at every point; endpoints use one-sided limits.
• Types of discontinuity:
  • Removable: limit exists but ≠ f(a) (or f(a) undefined); e.g. (x²−1)/(x−1) at x=1.
  • Jump: one-sided limits exist but differ; e.g. [x] at each integer.
  • Infinite: limit is infinite; e.g. 1/x at x=0.
• Example: f(x)=x² is continuous everywhere since lim_{x→a} x² = a² = f(a).
• Standard continuous families: polynomials (all ℝ), sin x, cos x (all ℝ), eˣ (all ℝ), rational functions except where denominator is zero, log x on (0,∞).

B. algebra of continuous functions

If f and g are continuous at a, their arithmetic combinations are too.

• Sum/difference: f ± g continuous at a.
• Product: f · g continuous at a.
• Scalar multiple: α f continuous at a.
• Quotient: f/g continuous at a, provided g(a) ≠ 0.
• Composition: if g is continuous at a and f is continuous at g(a), then f∘g is continuous at a.
• Reasoning basis: each rule follows from the matching limit law in III.B applied at a, since continuity means the limit equals the value.
• Example: h(x) = (x² + 1)·sin x is continuous on all of ℝ, being a product of the polynomial x²+1 and the continuous function sin x.
• Quotient caution: tan x = sin x / cos x is continuous everywhere except where cos x = 0, i.e. x = π/2 + nπ.