\( \begin{array}{l}\text { Discuss the continuity of } \\ \text { the function at } x=1\end{array} f(x)=\left\{\left.\begin{array}{ll}3 x-1 & \text { if } x<1 \\ 4 & \text { if } x=1 \\ 2 x & \text { if } x>1\end{array} \right\rvert\,\right. \)

To determine the continuity of the function \( f(x) \) at \( x = 1 \), let's analyze each aspect required for continuity. ### Given Function: \[ f(x) = \begin{cases} 3x - 1 & \text{if } x < 1 \\ 4 & \text{if } x = 1 \\ 2x & \text{if } x > 1 \end{cases} \] ### Conditions for Continuity at \( x = 1 \): A function \( f(x) \) is continuous at \( x = a \) if all the following conditions are met: 1. **\( f(a) \) is defined.** 2. **The limit \( \lim_{x \to a} f(x) \) exists.** 3. **\( \lim_{x \to a} f(x) = f(a) \).** Let's verify each condition for \( x = 1 \). #### 1. \( f(1) \) is Defined: From the function definition, \[ f(1) = 4. \] Thus, \( f(1) \) exists. #### 2. The Limit \( \lim_{x \to 1} f(x) \) Exists: We need to evaluate the left-hand limit (as \( x \) approaches 1 from below) and the right-hand limit (as \( x \) approaches 1 from above). - **Left-Hand Limit (\( x \to 1^- \)):** \[ \lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} (3x - 1) = 3(1) - 1 = 2. \] - **Right-Hand Limit (\( x \to 1^+ \)):** \[ \lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (2x) = 2(1) = 2. \] Since both one-sided limits are equal, \[ \lim_{x \to 1} f(x) = 2. \] #### 3. Comparing the Limit to \( f(1) \): \[ \lim_{x \to 1} f(x) = 2 \quad \text{and} \quad f(1) = 4. \] Since \( 2 \neq 4 \), the third condition for continuity is not satisfied. ### Conclusion: The function \( f(x) \) **is not continuous** at \( x = 1 \). Specifically, there is a **jump discontinuity** at this point because the limit of \( f(x) \) as \( x \) approaches 1 is 2, which does not equal the function's value at 1, which is 4. ### Visual Representation: A graph of \( f(x) \) around \( x = 1 \) would show: - **For \( x < 1 \):** A straight line with slope 3, passing through points like \( (0, -1) \) and \( (1, 2) \). - **At \( x = 1 \):** A distinct point at \( (1, 4) \). - **For \( x > 1 \):** A straight line with slope 2, starting from \( (1, 2) \) onwards. The sudden jump from the limit value 2 to the function value 4 at \( x = 1 \) illustrates the discontinuity.