# Pointwise versus uniform convergence > The cutoff N may depend on both ε and the point x. Course: [Real analysis: limits and rigorous reasoning](https://studybrix.com/en/courses/real-analysis/index.md) Level: advanced Prerequisites: Sequence limits and supremum norms. ## Notes ### Pointwise The cutoff N may depend on both ε and the point x. ### Uniform One N must work for all points in the domain at once. ### Supremum norm Uniform convergence is equivalent to sup_x abs(f_n(x)−f(x)) tending to zero. ### Continuity A uniform limit of continuous functions is continuous; pointwise limits need not be. ## Practice questions ### 1. Find the pointwise limit of f_n(x)=x^n on [0,1]. Sample answer / marking guidance: The limit is 0 for 0≤x<1 and 1 at x=1. Explanation: Credit both the interior and the endpoint. ### 2. Explain why x^n does not converge uniformly on [0,1]. Sample answer / marking guidance: Each x^n is continuous but the pointwise limit jumps at 1. A uniform limit of continuous functions would be continuous; equivalently the supremum error remains 1. Explanation: Credit either the continuity theorem with hypotheses or a valid supremum-error argument. ### 3. For f_n(x)=x/n on [0,2], find sup abs(f_n) when n=8. Answer: 0.25 Explanation: The maximum occurs at x=2 and equals 2/8. ### 4. For uniform convergence, can the cutoff N depend on x? A. Only when x is rational B. Yes, without restriction C. Only at positive x D. No Answer: No Explanation: A single cutoff must work across the whole domain. ### 5. Show that x/n converges uniformly to zero on [0,2]. Sample answer / marking guidance: The supremum error is 2/n, which tends to zero. Given ε>0, choose N>2/ε. Explanation: Credit a domain-wide bound. ### 6. Does pointwise convergence of continuous functions preserve continuity automatically? A. No B. Only because each term is bounded C. Yes in every domain D. The limit must be a polynomial Answer: No Explanation: The x^n example on [0,1] disproves the claim. ### 7. For a fixed 0