Show that the function fx - 3 - x- if x 1- is continuous at x - 1

lim_h → 0 f(1 h) = 3 1 + h = 2 + h = 2 lim_h → 0 f(1 #160;+ h) = #160;1 + #160;1 + h = 2 + h = 2 ⇒lim_→ 0 f(1 h) = f(1) = lim_h → 0 f ...

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Theorems for Continuity

Show that the...

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Show that the function $f (x) = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} 3 - x, & i f & x < 1 2, & i f & x = 1 1 + x, & i f & x > 1 \end{matrix}$ is continuous at $x = 1$

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f (x) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ \begin{matrix} (1 + a x)^{\frac{1}{x}}, if x < 0 b, if x = 0 \frac{(x + c)^{\frac{1}{3}} - 1}{(x + 1)^{\frac{1}{2}} - 1}, if x > 0 \end{matrix}

x = 0

f

f (x) =

{\begin{matrix} x, i f x \leq 1 5, i f x > 1 \end{matrix}

x = 0

x = 1

x = 2

a

b

f (x) =

⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} 3 a x + b, i f x \geq 1 11, i f x = 1 5 a x - 2 b, i f x < 1 \end{matrix} ⎫ ⎪ ⎬ ⎪ ⎭

x = 1

Is the function f defined by

continuous at x = 0? At x = 1? At x = 2?

f (x) = x, i f x \leq 1

5, i f x \geq 1

f (x)

x = 0 ? x = 1 ? x = 2 ?

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