Study Continuity and Differentiability of a Piecewise Function f(x)

Math Problem Statement

Study the behavior of the piecewise function f(x) defined as f(x) = e^x for x < 1 and f(x) = x - 1 for x >= 1. Analyze the continuity and differentiability of the function at x = 1 and calculate the limits as x approaches 1.

Solution

In this image, I can see a mathematical exercise written in Arabic. It appears to focus on a piecewise-defined function $f(x)$ over the real numbers $\mathbb{R}$ , and there are several tasks associated with this function. Here's the breakdown:

Function Definition:

The function $f(x)$ is defined piecewise as:

\begin{cases} e^x & \text{if } x < 1 \\ x - 1 & \text{if } x \geq 1 \end{cases}$$ ### Tasks: There are several questions related to this function: 1. **Continuity of $$f(x)$$**: To study the continuity of the function $$f(x)$$, likely at the point $$x = 1$$ (since that's where the piecewise definition changes). 2. **Limit of $$f(x)$$ as $$x \to 1$$**: Evaluate the left-hand and right-hand limits of $$f(x)$$ at $$x = 1$$. 3. **Expression of the function** $$f'(x)$$: Differentiability, likely asking for the derivative of the function. #### General Outline: 1. **Continuity**: The function is continuous if $$\lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) = f(1)$$. 2. **Differentiability**: For differentiability, we would also need to ensure the derivative exists and is the same on both sides at $$x = 1$$. Would you like me to go through solving each step in detail? Also, here are five follow-up questions to consider: 1. How is the derivative of a piecewise function evaluated? 2. What does the concept of continuity imply in real analysis? 3. How do we interpret limits from the left and right side at a point? 4. Why is differentiability stricter than continuity? 5. What role does exponential function $$e^x$$ play in this context? ### Tip: Always check continuity before attempting to verify differentiability, as discontinuity implies non-differentiability at that point.

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Math Problem Analysis

Mathematical Concepts

Piecewise Function
Continuity
Limits
Differentiability

Formulas

f(x) = e^x for x < 1
f(x) = x - 1 for x >= 1
Continuity: lim_{x → 1^-} f(x) = lim_{x → 1^+} f(x) = f(1)

Theorems

Continuity Theorem
Definition of Limits
Differentiability Condition

Suitable Grade Level

Grades 10-12

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