Converging Functions

Convergence of functions is introduced in Sequence of Functions | Real Analysis.

Uniform Convergence Theorem

Let $f_n$ be a sequence of Riemann integrable functions on $[a,b]$. Suppose $f_n$ converges to $f$ uniformly. Then $f$ is Riemann integrable on $[a,b]$ and $\forall x \in [a,b]$:

$$\int_a^x f_n(x)\,\text{d}x \text{ converges to } \int_a^x f(x)\,\text{d}x \text{ uniformly }$$

and:

$$\lim_{n\to \infty} \int_a^b f_n(x)\,\text{d}x = \int_a^b \lim_{n\to \infty} f_n(x)\,\text{d}x = \int_a^b f(x)\,\text{d}x$$

Proof Hint

• Consider $\frac{\epsilon}{2(b-a)}$ in place of $\epsilon$.
• Consider Cauchy criteria for $f_N$.
• Prove $f-f_N$ is Riemann integrable using Cauchy criteria.
• $f$ is Riemann integrable as $f=f_N + (f - f_N)$.

When $f_n$ converges to $f$ pointwise, it is not certain whether $f$ is Riemann integrable or not. An example where $f$ is not Riemann integrable:

$$\lim_{n\to\infty}u_n= \begin{cases} \;1 & x=q_k\;\text{where}\; k \le n\\ \;0 & \text{otherwise}\\ \end{cases}$$

Here $q_k$ is the enumeration of rational numbers in $[0,1]$.

Dominated Convergence Theorem

Let $f_n$ be a sequence of Riemann integrable functions on $[a,b]$. Suppose $f_n$ converges to $f$ pointwise where $f$ is Riemann integrable on $[a,b]$. If $\exists M>0\;\forall n\; \forall x \in [a,b]\;\text{s.t.}\; \lvert f_n(x) \rvert \le M$:

$$\lim_{n\to \infty} \int_a^b f_n(x)\,\text{d}x = \int_a^b f(x)\,\text{d}x$$

Monotone Convergence Theorem

Let $f_n$ be a sequence of Riemann integrable functions on $[a,b]$, and they are monotone (all increasing or decreasing, like $f_1 \le f_2 \cdots \le f_n$). Suppose $f_n$ converges to $f$ pointwise where $f$ is Riemann integrable on $[a,b]$. If $\exists M>0\;\forall n\; \forall x \in [a,b]\;\text{s.t.}\; \lvert f_n(x) \rvert \le M$:

$$\lim_{n\to \infty} \int_a^b f_n(x)\,\text{d}x = \int_a^b f(x)\,\text{d}x$$

Can be proven from the dominated convergence theorem.