Continuity and Differentiability

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To check continuity at x = 1, verify that lim(x→1) f(x) = f(1). Step 1: Find f(1) = 2(1) + 3 = 5 Step 2: Find lim(x→1) f(x) = lim(x→1) (2x + 3) = 2(1) + 3 = 5 Step 3: Compare: lim(x→1) f(x) = 5 = f…

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For f(x) = |x| = {-x if x < 0; x if x ≥ 0}, check continuity at x = 0. Step 1: Find f(0) = |0| = 0 Step 2: Find left-hand limit: lim(x→0⁻) f(x) = lim(x→0⁻) (-x) = 0 Step 3: Find right-hand limit: l…

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For f(x) = [x], analyze at each integer point in [0, 3). At x = 0: - Left limit: lim(x→0⁻) [x] doesn't exist in domain - Right limit: lim(x→0⁺) [x] = 0 - f(0) = 0 → Continuous At x = 1: - Left limit…

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For continuity at x = 2, we need lim(x→2⁻) f(x) = lim(x→2⁺) f(x) = f(2). Step 1: Find left-hand limit: lim(x→2⁻) f(x) = lim(x→2⁻) kx² = k(2)² = 4k Step 2: Find right-hand limit: lim(x→2⁺) f(x) = 3 (…

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Use the continuity definition when asked to verify or prove a function is continuous at a point c. The THREE CONDITIONS that must ALL be satisfied: 1. f(c) must be defined (the function exists at x …

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To show non-differentiability, prove the left and right derivatives are not equal. For f(x) = |x| = {-x if x < 0; x if x ≥ 0} Step 1: Find left-hand derivative at x = 0: f'(0⁻) = lim(h→0⁻) [f(0+h) -…

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THEOREM: If f is differentiable at c, then f is continuous at c. Important understanding: - Differentiability IMPLIES continuity - Continuous ≠ Differentiable (converse is FALSE) In symbols: f'(c) e…

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The Chain Rule: If f = v∘u, then df/dx = (dv/dt)(dt/dx) where t = u(x). For f(x) = sin(x²): Step 1: Identify composition - Inner function: u(x) = x² - Outer function: v(t) = sin(t) - Therefore: f(x)…