After introducing sequences and gaining some knowledge of some of their properties (I, II, III, and IV) we are ready to embark on the study of real analysis.


Physics is expressed best and most powerfully in the language of mathematics and a very useful mathematical concept for physics is the concept of a function.

Generally speaking a function is an association (it transforms an input signal from the first set into an output signal of a second set) between the elements of two sets.

The sequences we studied are a special case of functions: they take natural numbers (or a subset of them) as their input signals and map them to real numbers.

Now, more formally we introduce:

Definition 23 A function is a mapping between a set of real numbers to another set of real numbers $ \displaystyle f:D\subset \mathbb{R} \rightarrow \mathbb{R} \ \ \ \ \ (15)$ The set $ {D}$ is called the domain of the function The set of values taken by the output signals is called th range of the function. We represent the output signal by $ {f(x)}$, thus the former can be written as: $ {\left\lbrace f(x):x \in D \right\rbrace = f\left[ D \right] }$.

Sometimes we may not be interested in how the function maps the whole of $ {D}$ but just on a particular subset of $ {D}$. So it makes sense to introduce:

Definition 24 Given $ {E \subset D}$ it is $ {f\left[ E \right] = \left\lbrace f(x):x \in E \right\rbrace }$ is the image of $ {f}$ by $ {E}$.

As we did for sequences we can too define what is a bounded above function, a bounded below function, a bounded function and etc.

As an example we'll give:

Definition 25 $ {f}$ is said to be bounded iff $ {\exists \, \alpha > 0 : |f(x)| \leq \alpha \forall x \in D }$

We will now introduce some light topological notions in order to shed some light into the study of limits and continuity.

Definition 26 Given $ {E \subset \mathbb{R}}$ we'll say that $ {c \in \overline{\mathbb{R}}}$ is a limit point of $ {E}$ if there exists a sequence $ {x_n}$ of points in $ {E \setminus \left\lbrace c \right\rbrace }$ such as $ {\lim x_n = c}$. The set of limit points of $ {E}$ will be represented by $ {E^\prime}$. The set of points of $ {E}$ that aren't limit points will be called isolated points.

Once again so that we don't let things get too abstract let us give an example:

$ \displaystyle E = \left] 0,1\right[ \cup \left\lbrace 2 \right\rbrace $

It is easy to see (and we won't give a rigorous proof of that) that $ {E^\prime= \left[ 0,1 \right] }$ and that $ {2}$ is the only isolated point of $ {E}$.

Definition 27 We'll use the symbol $ {\displaystyle \lim _{x \rightarrow c^+}}$ to denote approximation to $ {c}$ by real numbers bigger than $ {c}$. In an analogous way we can also define $ {\displaystyle \lim _{x \rightarrow c^-}}$. Thus, we define $ {\displaystyle \lim _{x \rightarrow c^+} f(x) = a}$ if for all $ {x_n \in D}$ such as $ {x_n \rightarrow c^+}$ corresponds a sequence $ {f(x_n)}$ such as $ {f(x_n) \rightarrow a}$.

Definition 28 The symbol $ {D_{c^+}}$ will be used to denote $ {D \cap \left] c, \infty \right[ }$ and the symbol $ {D_{c^-}}$ will denote $ {D \cap \left] - \infty , c \right[ }$

As an example let us calculate

$ \displaystyle \lim _{x \rightarrow 0^+} \frac{1}{x} $

In this case it is $ {D_{0^+} = \left] 0, \infty \right[ }$ and $ {0^+ \in D^\prime_{c^+}}$ so that the limit we intend to calculate indeed makes sense.

If $ {x_n}$ is a sequence of points in $ {D^\prime_{c^+}}$ such as $ {x_n \rightarrow 0^+}$ then it follows that $ {\lim f(x_n)=\lim \dfrac{1}{x_n}=\dfrac{1}{0^+}=+\infty }$

Theorem 28 Given $ {D \subset \mathbb{R}}$, $ {f : D \rightarrow \mathbb{R}}$, $ {c \in D^\prime}$ let us suppose that $ {\displaystyle \lim_{x \rightarrow c} f(x) = a}$. Then, if $ {c \in D^\prime_{c^+}}$ it also is $ {\displaystyle \lim_{x \rightarrow c^+} f(x) = a }$. If $ {c \in D^\prime_{c^-}}$ it also is $ {\displaystyle \lim_{x \rightarrow c^-} f(x) = a }$. Proof: Let $ {x_n}$ be a sequence of points in $ {D_{c^+}}$ such as $ {x_n \rightarrow c}$. Since $ {x_n}$ is a sequence of points in $ {D \setminus \left\lbrace c \right\rbrace }$ (by our choice of $ {x_n}$) and $ {\displaystyle \lim_{x \rightarrow c} f(x) = a}$ (by hypothesis of the theorem) it follows from the definition of limit that $ { \lim f(x_n)= a}$. But this is just $ {\displaystyle \lim_{x \rightarrow c^+} = a}$ by definition. The case $ {\displaystyle \lim_{x \rightarrow c^-}}$ is proven with the same kind of reasoning. $ \Box$

As an application of theorem 28 let us calculate the following limit

$ \displaystyle \lim_{x \rightarrow 0} \dfrac{1}{x} $

It is easy to see that this limit doesn't exist. Let $ {f(x)=\dfrac{1}{x}}$ it is $ {\displaystyle \lim_{x \rightarrow 0^+} f(x) = +\infty}$ and $ {\displaystyle \lim_{x \rightarrow 0^-} f(x) = -\infty}$.

Since the limit from the left is different from the limit from the right we can conclude that $ {\displaystyle\lim_{x \rightarrow 0}\dfrac{1}{x}}$ doesn't exist.

Definition 29 $ { +\infty }$ is a limit point of $ {E}$ if $ {E}$ isn't bounded above in $ { \mathbb{R} }$. $ { -\infty }$ is a limit point of $ {E}$ if $ {E}$ isn't bounded below in $ {\mathbb{R}}$.

If you're having trouble understanding these definitions just think that if $ {E}$ isn't bounded above than it means that $ { \exists x_n \in E: \quad \lim x_n = +\infty }$.

Which is just the definition of limit point.

Definition 30 $ {c}$ is said to be a limit point of $ {E}$ if $ \displaystyle \forall \delta > 0 \; V(c,\delta) \cap E \setminus \left\lbrace c \right\rbrace \neq \emptyset$

Definition 31 Let $ {D \subset \mathbb{R} }$, $ {f : D \rightarrow \mathbb{R}}$, $ {c \in D^\prime}$ and $ { a \in \mathbb{R} }$. $ {f}$ has limit $ {a}$ in point $ {c}$ if for all sequences $ {x_n \in D \setminus \left\lbrace c \right\rbrace }$ such as $ {\lim x_n = c}$ it follows that $ {\lim f(x_n) = a}$.

We'll only define the limit of a function in limit points of the domain. Notice that by this way we can too define the limit of points that don't belong in the domain of the function.

As always a few examples will be provided in order for us to test our knowledge.

• Calculate the limit of $ {\displaystyle \lim_{x \rightarrow + \infty} \dfrac{1}{x} }$.
  $ { D = \mathbb{R} \setminus \left\lbrace 0 \right\rbrace }$ and $ { + \infty \in D^\prime }$ since $ {D}$ isn't bounded above in $ { \mathbb{R} }$. Thus the limit we set ourselves to calculate makes sense in our theory of limits.
  
  Let $ {x_n}$ be a sequence of points in $ {D}$ such as $ { x_n \rightarrow + \infty }$ and $ {f(x)=\dfrac{1}{x}}$, then $ {f(x_n)=\dfrac{1}{x_n}}$ and it always is $ {\lim f(x_n)=0}$.
  
  
• Calculate the limit of $ {\displaystyle \lim_{x \rightarrow + \infty} \sin x }$
  Choosing $ {f(x)= \sin x}$ we see that the domain is $ {D = \mathbb{R}}$. Thus $ {+\infty \in D^\prime}$
  
  Let us choose $ {x_n = n \pi}$. Thus $ {x_n \rightarrow +\infty }$ and $ {f(x_n)=\sin x_n = 0}$.
  
  In this case it trivially is $ {\lim f(x_n)=0}$.
  
  Now if we choose $ {y_n=\pi/2 + 2n\pi}$ it also is $ {y_n \rightarrow + \infty}$ but $ {f(y_n)= \sin (\pi/2+2n\pi)=1}$ and so $ {\lim f(y_n)=1}$.
  
  Thus we were able to find $ {x_n}$, $ {y_n}$ such as $ {\lim x_n = \lim y_n = + \infty}$ but $ {\lim f(x_n) \neq \lim f(y_n)}$. Thus $ {\displaystyle \lim_{x \rightarrow +\infty} \sin x }$ doesn't exist.
  
  

In order for us to proceed deeper in the study of limits and continuity we'll introduce the notions of one-sided limit. We'll use the symbols $ {\displaystyle \lim_{x \rightarrow c^+}}$ to denote approximation to $ {c}$ by real numbers that are bigger than $ {c}$. In an analogous way we can also define $ {\displaystyle \lim_{x \rightarrow c^-}}$ to denote the approximation to $ {c}$ by real numbers that are smaller than $ {c}$.

Formalizing the previous notions:

Definition 32 We'll say that $ {\displaystyle \lim_{x \rightarrow c^+} f(x)=a}$ if for all $ {x_n \in D}$ such as $ { x_n \rightarrow c^+}$ corresponds a sequence $ {f(x_n)}$ such as $ {f(x_n) \rightarrow a}$. The symbols $ {D_{c^+}}$ will be used to denote $ {D \cap \left] c, +\infty \right[ }$ and the symbols $ {D_{c^-}}$ will denote $ {D \cup \left] -\infty, c \right[ }$. The definitions of $ {\displaystyle \lim_{x \rightarrow c^-} f(x)=a}$ and $ {D_{c^-}}$ are done analogously.