[br][center][b]Continuidad[/b][/center]Una función [b][i]f(x)[/i][/b] es continua en un punto [b][i]x = a[/i][/b] si no hay “saltos”, “huecos” o “interrupciones” en ese punto. Intuitivamente, puedes dibujar la gráfica sin levantar el lápiz del papel. Una función [b][i]f(x)[/i][/b] es continua en [b][i]x = a[/i][/b] si se cumplen las tres condiciones:[br]a.      [b][i]F(a)[/i][/b] está definida.b.  Existe el límite[img width=63,height=28]data:image/png;base64,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[/img] c.   [img width=108,height=28]data:image/png;base64,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[/img][center][br][b]Continuidad en un intervalo[/b][br][/center][justify]Una función es continua en un intervalo si es continua en todos los puntos del intervalo. Si es continua en todos los números reales, decimos que es continua en ℝ. [b]Por ejemplo:[/b][br][br]Considera la función [b][i]f(x) = x².[br][/i][/b]Verificamos la continuidad en [b][i]x = 2:[br][br][/i][/b][/justify][list=1][*][b][i]  [b][i] F(2) = 4 [/i][/b]→ Está definida.[br][/i][/b][/*][*][b][i] [img width=89,height=28]data:image/png;base64,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[/img][/i][/b][/*][*][b][i] [img width=138,height=28]data:image/png;base64,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[/img][/i][/b][/*][/list][justify]Es decir, [b][i]f(x) [/i][/b][b][i]= x²[/i][/b] es continua en [b][i]x = 2[/i][/b] y en todos los números reales.[/justify][justify][b][br]Ejemplo [/b][b]de función no continua[/b][/justify][justify]Definimos la función por partes:[/justify][justify][img width=190,height=57]data:image/png;base64,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[/img][/justify][justify]Evaluamos la continuidad en [b][i]x = 1[/i][/b]:[/justify] 1.     [b][i]F (1) = 3[/i][/b] → Está definida.[justify]2.   [img width=206,height=28]data:image/png;base64,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[/img][/justify][justify]3.   [img width=169,height=30]data:image/png;base64,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[/img][/justify][justify][br]Como los resultados son distintos el límite no existe. Por ende, no es continua en [b][i]x [/i][/b][b][i]= 1.[/i][/b][/justify][justify][br][br][br][br][/justify][justify][br][br][br][/justify][br][br]