1. The problem is to understand the concept of continuous and discontinuous functions in mathematics. 2. A function $f(x)$ is said to be continuous at a point $x=a$ if the following three conditions are met: - $f(a)$ is defined. - The limit $\lim_{x \to a} f(x)$ exists. - The limit equals the function value: $\lim_{x \to a} f(x) = f(a)$. 3. If any of these conditions fail, the function is discontinuous at $x=a$. 4. There are different types of discontinuities: - **Jump discontinuity:** The left-hand and right-hand limits exist but are not equal. - **Infinite discontinuity:** The function approaches infinity near $x=a$. - **Removable discontinuity:** The limit exists but is not equal to the function value or the function is not defined at $x=a$. 5. Example: Consider the function $$f(x) = \begin{cases} x^2 & \text{if } x \neq 1 \\ 3 & \text{if } x = 1 \end{cases}$$ - Here, $f(1) = 3$. - The limit as $x \to 1$ is $\lim_{x \to 1} x^2 = 1$. - Since $\lim_{x \to 1} f(x) \neq f(1)$, the function is discontinuous at $x=1$ (removable discontinuity). 6. Summary: Continuity means no breaks, jumps, or holes in the graph of the function at the point considered. Final answer: A function is continuous at $x=a$ if $f(a)$ is defined, $\lim_{x \to a} f(x)$ exists, and both are equal; otherwise, it is discontinuous at $x=a$.