A rigorous analysis of functions of one variable.
A C- or better in (MATH 240, MATH 241, and MATH 310) or (MATH 340, MATH 341, and MATH 310)
The Real Numbers
The Completeness Axiom: The Natural, Rational,
and Irrational Numbers
Sequences of Real Numbers
The Convergence of Sequences
The Monotone Convergence Theorem, the Bolzano-Weierstrass
Theorem, and the Nested
Interval Theorem
Continuous Functions and Limits
Continuity
The Extreme Value Theorem
The Intermediate Value Theorem
Uniform Continuity
Limits
Differentiation
The Algebra of Derivatives
Differentiating Inverses and Compositions
The Lagrange Mean Value Theorem and
Its Geometric Consequences
The Cauchy Mean Value Theorem and Its Analytic Consequences
The Elementary Functions as Solutions of Differential Equations
The Natural Logarithm and the Exponential Functions
The Trigonometric Functions
The Inverse Trigonometric Functions
Integration
The Definition of the Integral and Criteria for Integrability
The First Fundamental Theorem of Calculus
The Convergence of Darboux Sums and Riemann Sums
Linearity, Monotonicity, and Additivity over Intervals
The Fundamental Theorems of Calculus and Their Consequences
The Second Fundamental Theorem of Calculus
The Existence of Solutions of Differential Equations
The Approximation of Integrals
Approximation by Taylor Polynomials
Taylor Polynomials and Order of Contact
The Lagrange Remainder Theorem
The Convergence of Taylor Polynomials
The Cauchy Integral Remainder Formula and
the Binomial Expansion
The Weierstrass Approximation Theorem
The Convergence of Sequences and Series of Functions
Sequences and Series of Numbers
Pointwise Convergences and Uniform Convergence
of Sequences of Functions
The Uniform Limit of Continuous Functions,
of Integrable Functions, and of
Differentiable Functions
Power Series