Tutorial: Limits: algebraic viewpoint

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This tutorial: Part A: Continuity from the graphical viewpoint

Go to Part B: Continuity from the algebraic viewpoint

(This topic is also in Section 10.3 in Finite Mathematics and Applied Calculus)

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Continuities and discontinuities

Let us look once again at the graph of the function $f$ we saw in the %%geometriclimitstut:

Notice that the graph of $f$ has "breaks" in it at two values of $x$ in its domain: one at $x = -1$, and the other at $x = 0$. We say that $f$ is discontinuous at such values of $x$. We say that $f$ is continuous at values of the its domain where the graph does not break, such as at $x = 1$. We can epress these ideas purely in terms of limits:

#[Continuous function][Función continua]#

Let $x = a$ be a point in the domain of the function $f.$ Then $f$ is %%contat $x = a$ if both of the following are true:

1. $\displaystyle \lim_{x \to a} f(x)$ %%exists.\t $\qquad$ \t So, if the left and right limits exist, they are equal. \\ 2. $\displaystyle \lim_{x \to a} f(x) = f(a)$ \t \t The limit equals the value of the function at the point $a$.

The function $f$ is said to be continuous on its domain if it is continuous at every point in its domain. If $f$ is not continuous at a particular point, $a$, in its domain, we say that $f$ is discontinuous at $x = a$ or that $f$ has a discontinuity at $x = a$.

#[Example][Ejemplo]#

#[In the graph we are studying:][En la gráfica que hemos sido estudiando:]#

%%Q #[Is $f$ continuous at $x = 1$?][¿Es continua $f$ en $x = 1$?]# %%A #[Check the definition:][Comprueba la definición:]#

1. $\displaystyle \lim_{x \to 1} f(x)$ #[exists and equals][existe y es igual a]# 0.5. ✓
2. $f(1) = 0.5$ #[as well.][también.]#✓

%%Therefore, $f$ %%is %%contat $x = 1.$

%%Q #[Is $f$ continuous at $x = 0$?][¿Es continua $f$ en $x = 0$?]#
%%A #[Check the definition:][Comprueba la definición:]#

1. $\displaystyle \lim_{x \to 0} f(x)$ #[does not exist][no existe]#.5. ✗

%%Therefore, $f$ %%is %%notcontat $x = 0.$

%%Q #[Is $f$ continuous at $x = -1$?][¿Es continua $f$ en $x = -1$?]#
%%A #[Check the definition:][Comprueba la definición:]#

1. $\displaystyle \lim_{x \to -1} f(x)$ #[exists, and equals 1][existe, y es igual a 1]#. ✓
2. $f(-1) = 0, $ #[which does not equal the limit.][que no es igual al límite.]#✗

%%Therefore, $f$ %%is %%notcontat $x = -1.$

Discontinuity versus singularity

#[Q][P]# What happens if a function $f$ is not defined at the isolated point $x = a$. Can we say that $f$ is automatically discontinuous there? For instance, $f(x) = 1/x$ is discontinuous at $x = 0$, right?
#[A][R]# Wrong. Continuity and discontinuity of a function are defined only for points in a function's domain; a function cannot be continuous at a point not in its domain, and it cannot be discontinuous there either. If a function is not defined at an isolated point $x = a$, we say that $f$ has a singularity at $x = a$. So, for exanple, $f(x) = 1/x$ has a singularity at $x = 0$ but we cannot say it is discontinuous there (even though the graph breaks at that point). Be aware that many authors use the term "discontinuity" to apply to singular points as well, but that is contrary to the accepted definition of the term.

Now try the exercises in Section 10.3 in Finite Mathematics and Applied Calculus. or move ahead to the next part of this tutorial by pressing "Next" on the sidebar.