Brunei Math Club

 Below we list some frequently used elementary functions. Algebraic functions Exponential functions Logarithm Trigonometric functions Inverse trigonometric functions Hyperbolic functions Algebraic functions A polynomial of $x$ $$f(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_{1}x+a_0$$ where $a_n,a_{n-1},\cdots ,a_0\in \mathbb{R}$ is a continuous function on $\mathbb{R}$. Such functions are called polynomial functions . If $g(x)$ and $h(x)$ are polynomial functions such that $h(x)\ne 0$, the function $f(x)$ defined by $$f(x)=\frac{g(x)}{h(x)}$$ is continuous on $\{x\mid x\in \mathbb{R},h(x)\ne 0\}$. Such functions are called rational functions . If $h(x)=1$ then $f(x)=g(x)$ so any polynomial functions are also rational functions (i.e., polynomial functions are a special case of rational functions). We can ``algebraically'' define functions other than polynomial or rational functions. For example, $f(x)=\sqrt{x}$ is not a rational function, bu...

 A function $f(x)$ is said to be continuous at $x=a$ if $f(x)$ converges to $f(a)$ as $x\to a$. Continuous functions are, in a sense, well-behaved and hence, easy to handle.  Definition (Continuous function) Let $f(x)$ be a function defined on an interval $I$ such that $a\in I$. The function $f(x)$ is said to be continuous at $a$ if $$\underset{x\to a}{lim}f(x)=f(a).$$ $f(x)$ is said to be a continuous function if it is continuous at every $x\in I$. Remark . According to the definition of limits, $f(x)$ is continuous at $x=a$ if the following condition is satisfied: For any $\epsilon >0$, there exists $\delta >0$ such that, for all $x\in \mathrm{dom}f$, $0<|x-a|<\delta$ implies $|f(x)-f(a)|<\epsilon$. □ Example . For the function $f(x)$ defined by $$f(x)=\{\begin{array}{cc}0& (x<0),\\ 1& (x\ge 0),\end{array}$$ $\underset{x\to 0}{lim}f(x)$ does n...