Properties of uniform limit f(x)

• categories: Real Analysis, Theorem

• Continuity: If a sequence of continuous functions $\{f_n(x)\}$ converges uniformly to $f(x)$, then $f(x)$ is also continuous.

• Integration: Uniform convergence allows for the interchange of limits and Riemann Integral:

$${\int }_a^b\underset{n\to \infty }{\lim }{f}_n(x)dx=\underset{n\to \infty }{\lim }{\int }_a^b{f}_n(x)dx$$