Real Analysis
Real Analysis tests limits, continuity, integration, and convergence in metric spaces. GATE MA favors counterexamples: pointwise versus uniform convergence, and when limit operations commute.
Core rules
- Completeness: every Cauchy sequence in R converges; compact subsets of R are closed and bounded.
- Uniform convergence preserves continuity; pointwise convergence does not.
- Weierstrass M-test: if |f_n(x)|≤M_n and ΣM_n converges, then Σf_n converges uniformly.
- Riemann integrability: a bounded f on [a,b] is integrable iff its set of discontinuities has measure zero.
- Interchange rules require uniform convergence: lim ∫ f_n = ∫ lim f_n under uniform convergence on a finite interval.
- Bolzano–Weierstrass: every bounded sequence in R has a convergent subsequence.
- Dini's theorem: a monotone sequence of continuous functions on a compact set converging pointwise to a continuous limit converges uniformly.
Worked example
Let f_n(x)=x^n on [0,1]. Pointwise, f_n(x)→0 for 0≤x<1 and f_n(1)=1, so the limit f is discontinuous. Since each f_n is continuous, convergence cannot be uniform: uniform limits of continuous functions are continuous. Alternatively, sup_{[0,1]}|x^n−f(x)|=1 for every n.
Traps
- Pointwise convergence does not allow swapping limits, derivatives, or integrals.
- A sequence of continuous functions can converge pointwise to a discontinuous function.
- Uniform continuity depends on the domain: x² is not uniformly continuous on R but is on [0,1].
- Compactness in metric spaces is sequential compactness; do not assume it in noncomplete spaces.