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Show that f (x) = sin(1/x) defined at (0, 1] is continuous.

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Answer to a math question Show that f (x) = sin(1/x) defined at (0, 1] is continuous.

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Rasheed
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Para demostrar que la función f(x) = \sin\left(\frac{1}{x}\right) definida en el intervalo (0,1] es continua, es importante recordar que la composición de funciones continuas es continua, así como que el seno es una función continua en todos los números reales.

Dado que el seno es una función continua en todos los números reales, la función g(x) = \frac{1}{x} es continua en el intervalo (0,1], ya que x es diferente de cero en dicho intervalo y \frac{1}{x} es continua para x > 0 .

Finalmente, al ser la función f(x) = \sin(g(x)) una composición de funciones continuas (el seno y g(x) ), se concluye que f(x) = \sin\left(\frac{1}{x}\right) es continua en el intervalo (0,1].

\textbf{Respuesta:} La función f(x) = \sin\left(\frac{1}{x}\right) definida en el intervalo (0,1] es continua.

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