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Calculus Level pending

If f ( x ) = x + { x } + x f(x) = x + \{-x \} + \lfloor x \rfloor , discuss the continuity of f [ 2 , 2 ] f \in [-2,2] .

Notation :

Discontinuous at all fractional values in [-2, 2] Can't say Discontinuous at all integral values in [-2, 2] Continuous at all integral values in [-2, 2]

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1 solution

Chew-Seong Cheong
Apr 25, 2017

f ( x ) = x + { x } + x Note that { u } = u u = x + ( x ) x + x = x x \begin{aligned} f(x) & = x + {\color{#3D99F6}\{-x\}} + \lfloor x \rfloor & \small \color{#3D99F6} \text{Note that }\{ u \} = u - \lfloor u \rfloor \\ & = x + {\color{#3D99F6}(- x) - \lfloor -x \rfloor} + \lfloor x \rfloor \\ & = \lfloor x \rfloor - \lfloor - x \rfloor \end{aligned}

f ( x ) = { 2 x for x is an integer. 2 x + 1 for x has fractional value. \implies f(x) = \begin{cases} 2 \lfloor x \rfloor & \small \color{#3D99F6} \text{for }x \text{ is an integer.} \\ 2 \lfloor x \rfloor + 1 & \small \color{#3D99F6} \text{for }x \text{ has fractional value.} \end{cases}

Therefore, f ( x ) = 2 x f(x) = 2 \lfloor x \rfloor is discontinuous at all integral values in [-2, 2] .

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