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Real analysis is a branch of mathematics that deals with the study of real numbers and their properties, such as continuity, convergence, differentiation, and integration. It provides a rigorous foundation for calculus and other mathematical topics, and it has many applications in science, engineering, economics, and other fields. The main focus of real analysis is on the concept of limits, which is used to study the behavior of functions and sequences. It also covers topics such as metric spaces, topology, and measure theory.

Let $S \subset \mathbb{R}$. A function $f: S \rightarrow \mathbb{R}$ is not continuous at $c \in S$ if

A function $f: S \rightarrow \mathbb{R}$ is not continuous at $c \in S$ if at least one of the following conditions hold:

- $\lim_{x \rightarrow c} f(x)$ does not exist.
- $\lim_{x \rightarrow c} f(x)$ exists, but it is not equal to $f(c)$.
- $f(c)$ is undefined (i.e., $c$ is not in the domain of $f$).

In other words, a function $f$ is not continuous at $c$ if it has a discontinuity at $c$. A discontinuity is a point at which the function fails to be continuous, and it can occur for various reasons, such as a jump, a hole, or an asymptote in the graph of the function.

Let $S \subset \mathbb{R}$. A function $f: S \rightarrow \mathbb{R}$ is not uniformly continuous on $S$ if

There are different equivalent ways to define uniform continuity, but one common definition is:

A function $f: S \rightarrow \mathbb{R}$ is not uniformly continuous on $S$ if there exists $\epsilon > 0$ such that for all $\delta > 0$, there exist $x, y \in S$ such that $|x – y| < \delta$ but $|f(x) – f(y)| \geq \epsilon$.

In other words, there is a positive distance $\epsilon$ such that no matter how small we choose a distance $\delta$ around any point in $S$, we can always find two points in $S$ that are closer than $\delta$ but whose function values differ by at least $\epsilon$. This means that the function is “not uniformly continuous” because the amount of variation in the function values depends on the location in $S$ and cannot be controlled by a single choice of $\delta$ for all points in $S$.

Intuitively, this means that the function can have arbitrarily rapid changes or oscillations, such that no matter how finely we try to sample or approximate it, there will always be some nearby points with very different function values. Alternatively, it may mean that the function has some kind of asymptotic behavior that prevents it from being “smoothly varying” or “locally approximable” at all points in $S$.

Let $S \subset \mathbb{R}$. A sequence of functions $f_n: S \rightarrow \mathbb{R}$ does not converge uniformly to $f: S \rightarrow \mathbb{R}$ if

There are a few equivalent ways to state the definition of uniform convergence, but one possible definition is:

A sequence of functions $f_n: S \rightarrow \mathbb{R}$ converges uniformly to $f: S \rightarrow \mathbb{R}$ if for every $\epsilon > 0$, there exists an $N \in \mathbb{N}$ such that for all $n \geq N$ and $x \in S$, we have $|f_n(x) – f(x)| < \epsilon$.

Therefore, a sequence of functions $f_n: S \rightarrow \mathbb{R}$ does not converge uniformly to $f: S \rightarrow \mathbb{R}$ if there exists an $\epsilon > 0$ such that for every $N \in \mathbb{N}$, there exists an $n \geq N$ and $x \in S$ such that $|f_n(x) – f(x)| \geq \epsilon$.

In other words, there is some fixed distance $\epsilon$ such that no matter how far out in the sequence you go, there will always be some point $x$ in $S$ and some function $f_n$ beyond that point that differ from $f(x)$ by at least $\epsilon$. This means that the sequence of functions does not converge uniformly to $f$.