List of Real Analysis Topics - Fundamental Theorems

Fundamental Theorems

• Monotone convergence theorem – relates monotonicity with convergence
• Intermediate value theorem – states that for each value between the least upper bound and greatest lower bound of the image of a continuous function there is at least one point in its domain that the function maps to that value
• Rolle's theorem – essentially states that a differentiable function which attains equal values at two distinct points must have a point somewhere between them where the first derivative is zero
• Mean value theorem – that given an arc of a differentiable curve, there is at least one point on that arc at which the derivative of the curve is equal to the "average" derivative of the arc
• Taylor's theorem – gives an approximation of a k times differentiable function around a given point by a k-th order Taylor-polynomial.
• L'Hôpital's rule – uses derivatives to help evaluate limits involving indeterminate forms
• Abel's theorem – relates the limit of a power series to the sum of its coefficients
• Lagrange inversion theorem – gives the taylor series of the inverse of an analytic function
• Darboux's theorem – states that all functions that result from the differentiation of other functions have the intermediate value property: the image of an interval is also an interval
• Heine–Borel theorem – sometimes used as the defining property of compactness
• Bolzano–Weierstrass theorem – states that each bounded sequence in Rn has a convergent subsequence.