1. Let's start by defining a continuous function. A function $f(x)$ is continuous at a point $x = a$ if all the following conditions are met: - $f(a)$ is defined. - The limit of $f(x)$ as $x$ approaches $a$ exists, i.e., $\lim_{x \to a} f(x)$ exists. - The limit equals the function value: $$\lim_{x \to a} f(x) = f(a).$$ 2. A function is continuous on an interval if it is continuous at every point in that interval. 3. Examples of continuous functions include: - Polynomial functions, such as $f(x) = x^2 + 3x + 1$. - Rational functions where the denominator is not zero, like $f(x) = \frac{1}{x - 2}$ (continuous everywhere except at $x=2$). - Exponential and logarithmic functions in their domains. - Trigonometric functions such as $\sin x, \cos x$. 4. Non-continuous functions have at least one point where one or more of the continuity conditions fail. 5. Types of discontinuities include: - Removable discontinuity: where the limit exists but is not equal to the function value or the function is not defined at that point. - Jump discontinuity: where the left-hand and right-hand limits exist but are not equal. - Infinite discontinuity: where the function approaches infinity near the point. 6. Examples of non-continuous functions: - The step function $f(x) = \begin{cases}1, & x<0 \\ 2, & x \ge 0\end{cases}$ has a jump discontinuity at $x=0$. - The function $f(x) = \frac{1}{x}$ has an infinite discontinuity at $x=0$. - A function with a hole, such as $f(x) = \frac{x^2 - 1}{x - 1}$, which simplifies to $f(x) = x+1$ except at $x=1$, where there is a removable discontinuity. Summary: - Continuous functions: polynomials, $\sin x, \cos x$, exponential functions, rational functions where denominator $\neq 0$. - Non-continuous functions: step functions, functions with holes, jump or infinite discontinuities.