Uniform Convergence: Criteria, Limit Theorems, and Function Series

All flashcards

Uniform convergence

For every $\varepsilon>0$, one index threshold makes $|f_n(x)-f(x)|<\varepsilon$ for every later $n$ and every $x$ in the domain.

Supremum-distance criterion

When the errors are bounded, $f_n$ converges uniformly to $f$ exactly when $\sup_{x\in E}|f_n(x)-f(x)|\to 0$.

Uniform Cauchy criterion

For real-valued functions, uniform convergence is equivalent to having, for every $\varepsilon>0$, one threshold after which $|f_m(x)-f_n(x)|<\varepsilon$ for all domain points and all later $m,n$.

Weierstrass M-test

If $|f_n(x)|\le M_n$ everywhere and $\sum_n M_n$ converges, then $\sum_n f_n$ converges absolutely and uniformly.

Derivative-limit theorem

On a closed interval, uniform convergence of $f_n'$ and convergence of $f_n$ at one point imply uniform convergence to a differentiable limit whose derivative is $\lim_n f_n'$.

Quiz questions

1. If continuous functions $f_n$ converge uniformly to $f$ on a domain, what must be true of $f$?Answer: The limit function is continuous on that domain.
2. Suppose Riemann-integrable functions $f_n$ converge uniformly to $f$ on a compact interval $I$. Which conclusion follows?Answer: The limit is Riemann integrable and its integral is the limit of the integrals.
3. For $f_n(x)=x^n$ on $[0,1]$, with limit $f(x)=0$ for $x<1$ and $f(1)=1$, why is convergence not uniform?Answer: For every $N$, some $x_N<1$ has $|f_N(x_N)-f(x_N)|=1/2$.
4. On $[a,b]$, which assumptions in the derivative-limit theorem ensure that $f_n$ converges uniformly to a differentiable function?Answer: The derivatives converge uniformly, and the functions converge at one point.
5. Under Dini's theorem, what conditions ensure uniform convergence on a compact space?Answer: Continuous functions increase monotonically to a continuous pointwise limit.