függvénysorozatok

Függvénysorozat refers to a sequence of functions, where each term in the sequence is a function itself. Mathematically, a sequence of functions is denoted as $(f_n)_{n=1}^{\infty}$, where $f_n: D \to \mathbb{R}$ for some domain $D$. This means that for each positive integer $n$, we have a function $f_n$ that maps elements from the set $D$ to real numbers.

The study of function sequences is primarily concerned with their convergence properties. Two fundamental types of convergence are pointwise convergence and uniform convergence. A function sequence $(f_n)$ is said to converge pointwise to a function $f$ on a set $D$ if for every $x \in D$, the sequence of real numbers $(f_n(x))_{n=1}^{\infty}$ converges to $f(x)$. This means that for each $x$ and each $\epsilon > 0$, there exists an $N$ (which may depend on $x$) such that for all $n > N$, $|f_n(x) - f(x)| < \epsilon$.

Uniform convergence is a stronger condition. A function sequence $(f_n)$ converges uniformly to a function $f$ on a set $D$ if for every $\epsilon > 0$, there exists an $N$ (independent of $x$) such that for all $n > N$ and for all $x \in D$, $|f_n(x) - f(x)| < \epsilon$. This independence of $N$ from $x$ is crucial. Uniform convergence has significant implications, as it preserves important properties like continuity and allows for term-by-term integration. If a sequence of continuous functions converges uniformly to a function $f$, then $f$ is also continuous.