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{{:((x)/(|x|)","if x lt 0 ),(-1"," if x ...

`{{:((x)/(|x|)","if x lt 0 ),(-1"," if x ge 0 ):}`

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f(0) = -1
`R.H.L = underset (x to 0^(+))(lim)f(x)`
` underset (h to 0)(lim)f(0 + h)= underset (h to 0)(lim)(-1)= -1`
L.H.L = ` underset (x to 0^(-))(lim)f(x)`
` underset (h to 0)(lim)f(0+h)=underset(h to 0)(lim)= (-h)/(|-h|)`
` underset (h to 0)(lim)(-h)/(h)= underset(h to 0)(lim)(-1)= -1`
`:'` R.H.L. = f(0) = L.H.L.
`:.`f(x) is continuous at x = 0
Therefore f(x) is always continous.
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NAGEEN PRAKASHAN-Continuity and Differentiability-Exercies 5.1
  1. f(x)={{:(|x|+3, if x le -3),(-2x, if -3 lt x lt 3 ),(6x + 2, if x ge3)...

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  2. {{:((|x|)/(x)","if x ne 0 ),(0"," if x = 0 ):}

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  3. {{:((x)/(|x|)","if x lt 0 ),(-1"," if x ge 0 ):}

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  4. f(x) {{:(x + 1"," if x ge 1 ),(x^(2)+ 1"," if x lt 1):}

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  5. f(x){{:(x^(3) - 3"," if x le 2 ),(x^(2) + 1"," if x gt 2 ):}

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  6. f(x){{:(10^(3) - 1"," if x le 1 ),(x^(2) "," if x gt 1 ):}

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  7. Is the function defined by f(x)={x+5, ifxlt=1x-5, ifx >1 a continuous...

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  8. Discuss the continuity of the function f, where f is defined by f(x){{...

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  9. f(x){{:(2x "," if x lt 0 ),(0"," if 0 le x le 1),(4x "," if x gt 1 ):}

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  10. f(x) {{:(-2"," if x le -1),(2x"," if -1lt x le 1),(2"," if x gt 1):}

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  11. Find the relationship between a and b so that the function f defined ...

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  12. For what value of lambdais the function defined by f(x)={lambda(x^2-2...

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  13. Show that the function defined by g(x)=x-[x] is discontinuous at all i...

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  14. Is the function defined by f(x)=x^2-sinx+5continuous at x=pi?

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  15. Discuss the continuity of the following functions:(a) f(x) = s in x + ...

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  16. Discuss the continuity of the cosine, cosecant, secant and cotangen...

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  17. Find all points of discontinuity of f, wheref(x)={(sinx)/x , ifx<0x+1,...

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  18. Determine if f defined byf(x)={x^2sin1/x , ifx!=0 0, ifx=0

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  19. Examine the continuity of f, where f is defined byf(x)={sinx-cosx , if...

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  20. f(x) = {{:((k cosx )/((pi - 2x)"," if x ne (pi)/(2))),(3"," if x = (p...

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