Question

Prove that every differentiable function is continuous. Examine continuity and differentiability of the function \[ f(x) = |x + 2| \text{at} x = -2. \]

Hint:

To check continuity, verify that the left-hand and right-hand limits at a point match the function value at that point. For differentiability, check that the left-hand and right-hand derivatives are equal.

Answer 1

We will first prove that every differentiable function is continuous, and then we will examine the continuity and differentiability of \( f(x) = |x + 2| \) at \( x = -2 \). 1. Step 1: Proving that every differentiable function is continuous If a function \( f(x) \) is differentiable at \( x = c \), it means that the derivative \( f'(c) \) exists, which implies that: \[ \lim_{x \to c} \frac{f(x) - f(c)}{x - c} \text{exists}. \] This implies that \( f(x) \) is continuous at \( x = c \), because: \[ \lim_{x \to c} f(x) = f(c) \] Thus, every differentiable function is continuous. 2. Step 2: Checking continuity of \( f(x) = |x + 2| \) at \( x = -2 \) To check the continuity of \( f(x) = |x + 2| \) at \( x = -2 \), we must check if: \[ \lim_{x \to -2} f(x) = f(-2) \] First, find \( f(-2) \): \[ f(-2) = |(-2) + 2| = |0| = 0 \] Now, check the one-sided limits: - For \( x \to -2^+ \), \( f(x) = x + 2 \), so: \[ \lim_{x \to -2^+} f(x) = \lim_{x \to -2^+} (x + 2) = 0 \] - For \( x \to -2^- \), \( f(x) = -(x + 2) \), so: \[ \lim_{x \to -2^-} f(x) = \lim_{x \to -2^-} -(x + 2) = 0 \] Since both one-sided limits are equal to \( 0 \), we conclude that: \[ \lim_{x \to -2} f(x) = f(-2) = 0 \] Thus, \( f(x) = |x + 2| \) is continuous at \( x = -2 \). 3. Step 3: Checking differentiability of \( f(x) = |x + 2| \) at \( x = -2 \) To check the differentiability at \( x = -2 \), we need to check if the left-hand and right-hand derivatives exist and are equal at \( x = -2 \). - For \( x > -2 \), \( f(x) = x + 2 \), so \( f'(x) = 1 \). - For \( x < -2 \), \( f(x) = -(x + 2) \), so \( f'(x) = -1 \). Since the left-hand and right-hand derivatives are not equal, \( f(x) = |x + 2| \) is not differentiable at \( x = -2 \). Final Answer: - The function \( f(x) = |x + 2| \) is continuous at \( x = -2 \). - The function \( f(x) = |x + 2| \) is not differentiable at \( x = -2 \).