Number of points of discontinuity of the func | vedclass

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(C) Given $f(x) = \sin(\{2^x + [2^x] + [3^{-x}]\})$.Since $[2^x]$ and $[3^{-x}]$ are integers,we use the property $\{x + n\} = \{x\}$ for any integer

Vedclass · Accepted Answer

$15$

Vedclass · Answer

(C) Given $f(x) = \sin(\{2^x + [2^x] + [3^{-x}]\})$.Since $[2^x]$ and $[3^{-x}]$ are integers,we use the property $\{x + n\} = \{x\}$ for any integer $n$.Thus,$f(x) = \sin(\{2^x + [2^x] + [3^{-x}]\}) = \sin(\{2^x + [3^{-x}]\}) = \sin(\{2^x\})$.The function $f(x) = \sin(\{2^x\})$ is discontinuous where $2^x$ is an integer,because the fractional part function $\{u\}$ is discontinuous at all integers $u$.For $x \in [0, 4]$,$2^x$ takes values in the interval $[2^0, 2^4] = [1, 16]$.The integers in the interval $[1, 16]$ are $1, 2, 3, \dots, 16$.However,$\{2^x\}$ is continuous at $x$ where $2^x$ is an integer if the limit from the left and right are equal.At $x=0$,$2^x=1$,$\{2^x\}$ jumps from $0$ to $0$ (as $x 	o 0^+$,$2^x 	o 1^+$,$\{2^x\} 	o 0$). Thus,it is continuous at $x=0$.For $x > 0$,$2^x = k$ (where $k \in \{2, 3, \dots, 16\}$),the function $\{2^x\}$ has a jump discontinuity.There are $15$ such values of $k$ $(k=2, 3, \dots, 16)$.Therefore,the number of points of discontinuity is $15$.

Number of points of discontinuity of the function $f(x) = \sin(\{2^x + [2^x] + [3^{-x}]\})$ for $x \in [0, 4]$ is (where $[.]$ and $\{.\}$ denote the greatest integer and fractional part functions,respectively).

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