Definition: A function $f$ is continuous at a number $a$ if
$$\lim_{x\to a}{f(x)}=f(a)$$
Notice that Definition implicitly requires three things if $f$ is continuous at $a$
1. f(a) is defined (that is, $a$ is in domain of $f$)
2. $\lim_{x\to a}{f(x)}$ exists
3. $\lim_{x\to a}{f(x)}=f(a)$
The definition says that $f$ is continuous at $a$ if $f(x)$ approaches $f(a)$ as x approaches a.
Right Continuity
A function $f$ is continuous from right at a number $a$ if
$$\lim_{x\to a^{+}}{f(x)}=f(a)$$
Left Continuity
A function $f$ is continuous from left at a number $a$ if
$$\lim_{x\to a^{-}}{f(x)}=f(a)$$
Continuous On An Interval
A function $f$ is continuous on an interval if it is continuous at every number in the interval. (If f is defined only on one side of an endpoint of the interval, we understand continuous at the endpoint to mean continuous from the right or continuous from the left.)
Continuity Laws
If $f$ and $g$ are continuous at $a$ and c is a constant, then the following functions are also continuous at $a$.
1. $f+g$
2. $f-g$
3. c$f$
4. $fg$
5. $\frac{f}{g}$, if $g(a)\neq 0$
Theorem
(a) Any polynomial is continuous everywhere that is, it is continuous on $\mathbb{R}=(-\infty, \infty)$.
(b) Any rational function is continuous wherever it is defined that is, it is continuous on its domain.
Remark:
The following types of functions are continuous at every number in their domains:
~Polynomials
~Rational functions
~Root functions
~Trigonometric functions
~Inverse trigonometric functions
~Exponential functions
~Logarithmic functions
Theorem
If $g$ is continuous at $a$ and $f$ is continuous at $g(a)$, then the composite function f0g is given by (f0g)(x)=f(g(x)) is continuous at $a$
The Intermediate Value Theorem
Suppose that $f$ is continuous on the closed interval $[a,b]$ and $N$ let be any number between $f(a)$ and $f(b)$, where $f(a)\neq f(b)$ . Then there exists a number $c$ in $(a,b)$ such that $f(c)= N$ .
$$\lim_{x\to a}{f(x)}=f(a)$$
Notice that Definition implicitly requires three things if $f$ is continuous at $a$
1. f(a) is defined (that is, $a$ is in domain of $f$)
2. $\lim_{x\to a}{f(x)}$ exists
3. $\lim_{x\to a}{f(x)}=f(a)$
The definition says that $f$ is continuous at $a$ if $f(x)$ approaches $f(a)$ as x approaches a.
Right Continuity
A function $f$ is continuous from right at a number $a$ if
$$\lim_{x\to a^{+}}{f(x)}=f(a)$$
Left Continuity
A function $f$ is continuous from left at a number $a$ if
$$\lim_{x\to a^{-}}{f(x)}=f(a)$$
Continuous On An Interval
A function $f$ is continuous on an interval if it is continuous at every number in the interval. (If f is defined only on one side of an endpoint of the interval, we understand continuous at the endpoint to mean continuous from the right or continuous from the left.)
Continuity Laws
If $f$ and $g$ are continuous at $a$ and c is a constant, then the following functions are also continuous at $a$.
1. $f+g$
2. $f-g$
3. c$f$
4. $fg$
5. $\frac{f}{g}$, if $g(a)\neq 0$
Theorem
(a) Any polynomial is continuous everywhere that is, it is continuous on $\mathbb{R}=(-\infty, \infty)$.
(b) Any rational function is continuous wherever it is defined that is, it is continuous on its domain.
Remark:
The following types of functions are continuous at every number in their domains:
~Polynomials
~Rational functions
~Root functions
~Trigonometric functions
~Inverse trigonometric functions
~Exponential functions
~Logarithmic functions
Theorem
If $g$ is continuous at $a$ and $f$ is continuous at $g(a)$, then the composite function f0g is given by (f0g)(x)=f(g(x)) is continuous at $a$
The Intermediate Value Theorem
Suppose that $f$ is continuous on the closed interval $[a,b]$ and $N$ let be any number between $f(a)$ and $f(b)$, where $f(a)\neq f(b)$ . Then there exists a number $c$ in $(a,b)$ such that $f(c)= N$ .