Math Problem Statement
Discuss the continuity of the piecewise function f(x) at x = 1, where:
f(x) =
x + 1, if x >= 1
2x, if x < 1.
Solution
The problem asks us to discuss the continuity of the piecewise function at , where:
\begin{cases} x + 1, & \text{if } x \geq 1, \\ 2x, & \text{if } x < 1. \end{cases}$$ To discuss continuity at $$x = 1$$, we must verify three conditions: 1. **Existence of $$f(1)$$:** From the definition of the function: $$f(1) = 1 + 1 = 2 \quad \text{(since $$x \geq 1$$)}.$$ 2. **Existence of the left-hand limit (LHL) as $$x \to 1^-$$:** For $$x < 1$$, $$f(x) = 2x$$. Therefore: $$\lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} 2x = 2(1) = 2.$$ 3. **Existence of the right-hand limit (RHL) as $$x \to 1^+$$:** For $$x \geq 1$$, $$f(x) = x + 1$$. Therefore: $$\lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (x + 1) = 1 + 1 = 2.$$ Since: - $$f(1) = 2$$, - $$\lim_{x \to 1^-} f(x) = 2$$, - $$\lim_{x \to 1^+} f(x) = 2$$, and the left-hand limit, right-hand limit, and $$f(1)$$ are all equal, $$f(x)$$ is **continuous at $$x = 1$$**. --- Would you like a detailed explanation of any step or concept? ### Related Questions: 1. How is the continuity of a piecewise function tested for other values of $$x$$? 2. Can a function be continuous but not differentiable at a point? 3. What are the implications of left-hand and right-hand limits not matching at a point? 4. Why is the definition of $$f(1)$$ crucial for continuity? 5. How does continuity differ from the concept of differentiability? ### Tip: Always check both the function's value and its limits when verifying continuity, as both are equally important.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Piecewise functions
Continuity
Limits
Formulas
Left-hand limit: lim(x → c^-) f(x)
Right-hand limit: lim(x → c^+) f(x)
Theorems
Definition of continuity at a point
Suitable Grade Level
Grades 11-12