Continuity of a Piecewise Function at x = 0

Math Problem Statement

Analyze the piecewise function f(x) defined as: f(x) = { x + 2, x < 0 3, x = 0 x^2 - 4, x > 0 }, where a = 0. Determine the continuity of f(x) at x = 0.

Solution

The piecewise function $f(x)$ given in the image is defined as:

\begin{cases} x + 2, & x < 0 \\ 3, & x = 0 \\ x^2 - 4, & x > 0 \end{cases}, \quad a = 0$$ This function describes the behavior of $$f(x)$$ across three intervals: 1. $$x < 0$$: $$f(x) = x + 2$$ 2. $$x = 0$$: $$f(x) = 3$$ 3. $$x > 0$$: $$f(x) = x^2 - 4$$ --- ### Analysis of the Function: #### 1. Left of $$x = 0$$ ( $$x < 0$$ ) For values less than $$0$$, the function is linear: $$f(x) = x + 2$$ - For instance: - If $$x = -1$$, then $$f(-1) = -1 + 2 = 1$$ - If $$x = -2$$, then $$f(-2) = -2 + 2 = 0$$ --- #### 2. At $$x = 0$$ The function explicitly states that $$f(0) = 3$$. --- #### 3. Right of $$x = 0$$ ( $$x > 0$$ ) For positive values, the function is quadratic: $$f(x) = x^2 - 4$$ - For instance: - If $$x = 1$$, then $$f(1) = 1^2 - 4 = -3$$ - If $$x = 2$$, then $$f(2) = 2^2 - 4 = 0$$ --- ### Continuity Check at $$x = 0$$ To check whether $$f(x)$$ is continuous at $$x = 0$$, we need the left-hand limit (LHL), right-hand limit (RHL), and the function value $$f(0)$$. - **Left-hand limit** ($$x \to 0^-$$): $$\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (x + 2) = 0 + 2 = 2$$ - **Right-hand limit** ($$x \to 0^+$$): $$\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} (x^2 - 4) = 0^2 - 4 = -4$$ - **Function value**: $$f(0) = 3$$ Since: $$\text{LHL} = 2, \quad \text{RHL} = -4, \quad \text{and} \quad f(0) = 3,$$ the left-hand limit, right-hand limit, and function value do not match. **Therefore, $$f(x)$$ is not continuous at $$x = 0$$.** --- ### Summary: - $$f(x)$$ is piecewise-defined with distinct behaviors in different intervals. - The function is **not continuous at $$x = 0$$** because the limits from the left and right differ, and neither matches $$f(0) = 3$$. --- ### Do you want me to: 1. Plot the graph of $$f(x)$$? 2. Explain the discontinuity more in-depth? 3. Find the derivative of $$f(x)$$ in each interval? 4. Check differentiability at $$x = 0$$? 5. Solve for specific values of $$x$$ for $$f(x) = k$$, where $$k$$ is a given constant? **Tip**: Always analyze piecewise functions by checking the limits and values carefully at the points where the definition changes.

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Math Problem Analysis

Mathematical Concepts

Piecewise Functions
Continuity
Limits

Formulas

Left-hand limit (LHL)
Right-hand limit (RHL)
Continuity condition: LHL = RHL = f(c)

Theorems

Definition of Continuity
Limit Laws

Suitable Grade Level

Grades 10-12

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