Foundations · Michaël PerreaultEssential Calculus Theorems

Foundations

Real Analysis Theorems

Michaël Perreault · 2025

Real analy­sis deals with the rig­or­ous study of real num­bers, se­quences, series, limits, con­ti­nu­ity, dif­fer­en­ti­a­tion, and in­te­gra­tion of real-valued func­tions. The the­o­rems in this sec­tion are fun­da­men­tal to un­der­stand­ing the prop­er­ties of func­tions on the real line.

  • In­ter­me­di­ate Value The­o­rem (IVT)
  • State­ment: If a func­tion ff is con­tin­u­ous on the closed in­ter­val [a,b][a, b], and kk is any number be­tween f(a)f(a) and f(b)f(b), then there exists at least one number cc in the open in­ter­val (a,b)(a, b) such that f(c)=kf(c) = k.
  • Sig­nif­i­cance: Guar­an­tees the ex­is­tence of a root or a spe­cif­ic value within an in­ter­val for con­tin­u­ous func­tions.
  • For­mu­la:
If f is continuous on [a,b] and f(a)≤k≤f(b) (or f(b)≤k≤f(a)), then ∃c∈(a,b) such that f(c)=k.\text{If } f \text{ is continuous on } [a, b] \text{ and } f(a) \le k \le f(b) \text{ (or } f(b) \le k \le f(a) \text{)}, \text{ then } \exists c \in (a, b) \text{ such that } f(c) = k.
  • Ex­treme Value The­o­rem (EVT)
  • State­ment: If a func­tion ff is con­tin­u­ous on the closed in­ter­val [a,b][a, b], then ff at­tains both a max­i­mum value and a min­i­mum value on [a,b][a, b].
  • Sig­nif­i­cance: En­sures that con­tin­u­ous func­tions on closed in­ter­vals have ab­solute ex­trema.
  • For­mu­la:
If f is continuous on [a,b], then ∃x1,x2∈[a,b] such that f(x1)≤f(x)≤f(x2) for all x∈[a,b].\text{If } f \text{ is continuous on } [a, b], \text{ then } \exists x_1, x_2 \in [a, b] \text{ such that } f(x_1) \le f(x) \le f(x_2) \text{ for all } x \in [a, b].
  • Mean Value The­o­rem (MVT)
  • State­ment: If a func­tion ff is con­tin­u­ous on the closed in­ter­val [a,b][a, b] and dif­fer­en­tiable on the open in­ter­val (a,b)(a, b), then there exists at least one number cc in (a,b)(a, b) such that
f′(c)=f(b)−f(a)b−af'(c) = \frac{f(b) - f(a)}{b - a}
  • Sig­nif­i­cance: Re­lates the av­er­age rate of change of a func­tion over an in­ter­val to its in­stan­ta­neous rate of change at some point within the in­ter­val.
  • For­mu­la:
If f is continuous on [a,b] and differentiable on (a,b), then ∃c∈(a,b) such that f′(c)=f(b)−f(a)b−a.\text{If } f \text{ is continuous on } [a, b] \text{ and differentiable on } (a, b), \text{ then } \exists c \in (a, b) \text{ such that } f'(c) = \frac{f(b) - f(a)}{b - a}.
If f is continuous on [a,b] and differentiable on (a,b), then ∃c∈(a,b) such that f′(c)=f(b)−f(a)b−a.\text{If } f \text{ is continuous on } [a, b] \text{ and differentiable on } (a, b), \text{ then } \exists c \in (a, b) \text{ such that } f'(c) = \frac{f(b) - f(a)}{b - a}.
  • Rolle's The­o­rem
  • State­ment: A spe­cial case of the MVT. If a func­tion ff is con­tin­u­ous on the closed in­ter­val [a,b][a, b], dif­fer­en­tiable on the open in­ter­val (a,b)(a, b), and f(a)=f(b)f(a) = f(b), then there exists at least one number cc in (a,b)(a, b) such that f′(c)=0f'(c) = 0.
  • Sig­nif­i­cance: Guar­an­tees the ex­is­tence of a crit­i­cal point be­tween two points where the func­tion has the same value.
  • For­mu­la:
If f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then ∃c∈(a,b) such that f′(c)=0.\text{If } f \text{ is continuous on } [a, b], \text{ differentiable on } (a, b), \text{ and } f(a) = f(b), \text{ then } \exists c \in (a, b) \text{ such that } f'(c) = 0.
  • Fun­da­men­tal The­o­rem of Cal­cu­lus (FTC)
    • State­ment: Con­nects dif­fer­en­ti­a­tion and in­te­gra­tion. It has two main parts:
      • Part 1: If ff is con­tin­u­ous on [a,b][a, b], then the func­tion gg de­fined by
g(x)=∫axf(t)dtg(x) = \int_a^x f(t) \, dt

is con­tin­u­ous on [a,b][a, b], dif­fer­en­tiable on (a,b)(a, b), and g′(x)=f(x)g'(x) = f(x) for all xx in (a,b)(a, b).

  • Part 2: If ff is con­tin­u­ous on [a,b][a, b], and FF is any an­ti­deriv­a­tive of ff on [a,b][a, b] (i.e., F′(x)=f(x)F'(x) = f(x)), then
∫abf(x)dx=F(b)−F(a)\int_a^b f(x) \, dx = F(b) - F(a)
  • Sig­nif­i­cance: Part 1 shows that dif­fer­en­ti­a­tion and in­te­gra­tion are in­verse op­er­a­tions. Part 2 pro­vides a method for eval­u­at­ing def­i­nite in­te­grals.
  • For­mu­las:
Part 1: ddx∫axf(t)dt=f(x)\text{Part 1: } \frac{d}{dx} \int_a^x f(t) \, dt = f(x)
Part 2: ∫abf(x)dx=F(b)−F(a), where F′(x)=f(x)\text{Part 2: } \int_a^b f(x) \, dx = F(b) - F(a), \text{ where } F'(x) = f(x)