🎓 Deeper Dive — Adults

Real analysis builds calculus rigorously from the real numbers.

Limit (ε-δ): lim x→a f(x) = L iff ∀ε>0 ∃δ>0 such that 0 < |x−a| < δ → |f(x) − L| < ε.

Continuity: f continuous at a iff lim x→a f(x) = f(a). Differentiability ⟹ continuity (but not converse).

Cauchy sequences: |aₙ − aₘ| → 0 as n,m → ∞. In ℝ, every Cauchy sequence converges (completeness).

Foundation for measure theory, probability, functional analysis, ODEs, PDEs.