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Real Analysis - Function Properties
39
Flashcards
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f(x) = |x|
Continuous everywhere, not differentiable at x = 0, no asymptotes.
f(x) = \tan(x)
Continuous for all x ≠ (2k+1)π/2, not differentiable at x = (2k+1)π/2, vertical asymptote at x = (2k+1)π/2.
f(x) = \cos^2(x)
Continuous everywhere, differentiable everywhere except at points where \sin(2x) = 0, no asymptotes.
f(x) = e^{\sin(x)}
Continuous everywhere, differentiable everywhere, no asymptotes.
f(x) = \tan(x) + \sec(x)
Continuous on (-\frac{\pi}{2}, \frac{\pi}{2}) and differentiable on (-\frac{\pi}{2}, \frac{\pi}{2}), vertical asymptotes at x = (2k+1)\frac{\pi}{2}, k \in \mathbb{Z}.
f(x) = \text{ln}(|x|)
Continuous for x ≠ 0, differentiable for x ≠ 0, vertical asymptote at x = 0.
f(x) = \sqrt{x^2 + 1}
Continuous everywhere, differentiable everywhere except at x = 0, no asymptotes.
f(x) = x^x for x > 0
Continuous for x > 0, differentiable for x > 0, no asymptotes.
f(x) = \frac{\text{ln}(x)}{x}
Continuous for x > 0, differentiable for x > 0, horizontal asymptote at y = 0.
f(x) = \exp(-x^2)
Continuous everywhere, differentiable everywhere, no asymptotes.
f(x) = \sqrt{x}
Continuous for x ≥ 0, differentiable for x > 0, no asymptotes.
f(x) = \frac{x^2 - 1}{x - 1}
Continuous for x ≠ 1, differentiable for x ≠ 1, removable discontinuity at x = 1.
f(x) = \sin(\frac{1}{x})
Discontinuous at x = 0, not differentiable at x = 0, oscillatory discontinuity at x = 0.
f(x) = \arcsin(x)
Continuous on [-1, 1], differentiable on (-1, 1), no asymptotes.
f(x) = |x|e^{-x}
Continuous everywhere, differentiable everywhere except at x = 0, no asymptotes.
f(x) = xe^{-1/x^2} for x ≠ 0, f(x) = 0 for x = 0
Continuous everywhere, not differentiable at x = 0, no asymptotes.
f(x) = \cos(x)
Continuous everywhere, differentiable everywhere, no asymptotes.
f(x) = \frac{sin(x)}{x}
Continuous for x ≠ 0, differentiable for x ≠ 0, no asymptotes.
f(x) = \text{sgn}(x)
Discontinuous at x = 0, not differentiable at x = 0, no asymptotes.
f(x) = \theta(x) - \text{Heaviside step function}
Discontinuous at x = 0, not differentiable at x = 0, no asymptotes.
f(x) = x |\sin(x)|
Continuous everywhere, not differentiable at multiples of π, no asymptotes.
f(x) = \text{ln}(x) + x^2
Continuous for x > 0, differentiable for x > 0, no asymptotes.
f(x) = x^3
Continuous everywhere, differentiable everywhere, no asymptotes.
f(x) = x^{2/3}
Continuous everywhere, not differentiable at x = 0, no asymptotes.
f(x) = \cos(\sqrt{x})
Continuous for x ≥ 0, differentiable for x > 0, no asymptotes.
f(x) = \frac{1}{x^2}
Continuous for x ≠ 0, not differentiable at x = 0, vertical asymptote at x = 0.
f(x) = \frac{1}{1 + e^{-x}}
Continuous everywhere, differentiable everywhere, horizontal asymptotes at y = 0 and y = 1.
f(x) = \frac{x}{1 + x^2}
Continuous everywhere, differentiable everywhere, horizontal asymptote at y = 0.
f(x) = \frac{1}{1 + x^2}
Continuous everywhere, differentiable everywhere, horizontal asymptote at y = 0.
f(x) = e^x
Continuous everywhere, differentiable everywhere, no asymptotes.
f(x) = \frac{1}{x}
Continuous for x ≠ 0, not differentiable at x = 0, vertical asymptote at x = 0.
f(x) = \frac{x}{|x|}
Discontinuous at x = 0, not differentiable at x = 0, no asymptotes.
f(x) = x^2 \sin(\frac{1}{x}) for x ≠ 0, f(x) = 0 for x = 0
Continuous everywhere, differentiable everywhere except at x = 0, no asymptotes.
f(x) = \ln(x)
Continuous for x > 0, differentiable for x > 0, vertical asymptote at x = 0.
f(x) = \sin(x)
Continuous everywhere, differentiable everywhere, no asymptotes.
f(x) = \arctan(x)
Continuous everywhere, differentiable everywhere, horizontal asymptotes at y = -\frac{\pi}{2} and y = \frac{\pi}{2}.
f(x) = \frac{x^2}{\sin(x)} for x ≠ n\pi, n\in\mathbb{Z}, f(x) = \lim_{y\to x} \frac{y^2}{\sin(y)} for x = n\pi
Continuous everywhere, differentiable everywhere except at x = nπ, no asymptotes.
f(x) = \ln(x^2 + 1)
Continuous everywhere, differentiable everywhere, no asymptotes.
f(x) = \frac{x}{\sqrt{x^2 + 1}}
Continuous everywhere, differentiable everywhere, no asymptotes.
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