Real analysis dives deep into the foundational elements that makes calculus possible, rigorously proving the "why" behind the "how." From the simple beauty of sequences to the intricate choreography of metric spaces, it defines the stage where calculus takes its most rigorous bow. ![[DALLE3_RealAnalysis.png]] ## Concept Tree ### Basic 1. **Limits**: Understand the ε-δ definition of limits. 2. **Continuity**: Grasp the definition and properties of continuous functions. 3. **Derivatives**: Familiarity with differentiation and basic properties. 4. **Sequences and Series**: Comprehend convergence, divergence, and the basic tests. 5. **Riemann Integration**: Grasp the partitioning technique and understand basic integrable functions. ### Advanced 1. **Uniform Continuity**: Move beyond basic continuity, understand where uniformity comes into play. 2. **Pointwise and Uniform Convergence**: Differentiating the two and understanding their applications. 3. **Differentiation Theorems**: Master the Mean Value Theorem, Rolle's Theorem, and Taylor's Theorem. 4. **Lebesgue Integration**: Transition from Riemann to Lebesgue's more general framework. 5. **Fourier Series**: Decompose functions into their sinusoidal components. ### Mastery 1. **Functional Analysis**: Navigate the landscape of Banach and Hilbert spaces. 2. **Measure Theory**: Get the nuts and bolts of abstract measure and integration. 3. **Complex Analysis Interface**: Understand the connections to complex variables and the complex plane. 4. **Distribution Theory**: Grasp the theory that generalizes functions and allows derivatives of discontinuous functions. 5. **Stochastic Processes Interface**: Comprehend the interaction between Real Analysis and probability theory, like Brownian motion.