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 \(x\) \[f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0\] 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 \(h(x) \neq 0\), the function \(f(x)\) defined by \[f(x) = \frac{g(x)}{h(x)}\] is continuous on \(\{x \mid x \in \mathbb{R}, h(x) \neq 0\}\). 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 \[\lim_{x\to a}f(x) = f(a).\] \(f(x)\) 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 \(\varepsilon > 0\), there exists \(\delta > 0\) such that, for all \(x \in \mathrm{dom}f\), \(0 < |x - a| < \delta\) implies \(|f(x) - f(a)| < \varepsilon\). □ Example . For the function \(f(x)\) defined by \[f(x) = \left\{ \begin{array}{cc} 0 & (x < 0),\\ 1 & (x \geq 0), \end{array}\right.\] \(\lim_{x \to 0}f(x)\) does not exist. Therefore