Construction of the field of real numbers and the least upper-bound property.
Review of sets, countable & uncountable sets. Metric Spaces: topological
properties, the topology of Euclidean space. Sequences and series. Continuity:
definition and basic theorems, uniform continuity, the Intermediate Value
Theorem. Differentiability on the real line: definition, the Mean Value
Theorem. The Riemann-Stieltjes integral: definition and examples, the
Fundamental Theorem of Calculus. Sequences and series of functions, uniform
convergence, the Weierstrass Approximation Theorem. Differentiability in higher
dimensions: motivations, the total derivative, and basic theorems. Partial
derivatives, characterization of continuously-differentiable functions. The
Inverse and Implicit Function Theorems. Higher-order derivatives.

Suggested books and references:

Rudin, W., Principles of Mathematical Analysis, McGraw-Hill, 1986.

Apostol, T. M., Mathematical Analysis, Narosa, 1987.