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#1 2007-09-16 19:20:01

John Ketchum
Guest

Continuity

Find the points at which f is discontinuous. At which of these points is f right or left continuous?
a)       { 2x+1  x<-1
    f(x) { 3x      [-1,1]
          { 2x-1   x>1

b)      { sqrt(-x)  x<0
   f(x) { 1           [0,1]
         { sqrt(x)   x>1

#2 2007-09-17 03:40:06

John E. Franklin
Member
Registered: 2005-08-29
Posts: 3,588

Re: Continuity

part "a" looks like a lightning bolt, but not too jagged.
It's all connected together in with two corners at about 165 degree
angles perhaps.   invtan(3) is angle in uniform rotating format such as radians or degrees.
Slope is not uniformly rotating because it is the division of rise over run.
For the m=2 parts on left and right in y=mx+b, it goes up twice as fast as it goes rightward.  And the invtan(2) is probably about 70 degrees, perhaps.
So the overall shape of the inner line segment with two half lines on the
sides looks like the letter "S", or a ligtning bolt, not a Z, that goes the other way.

Last edited by John E. Franklin (2007-09-17 03:44:58)


igloo myrtilles fourmis

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#3 2007-09-17 06:18:31

HallsofIvy
Guest

Re: Continuity

John Ketchum wrote:

Find the points at which f is discontinuous. At which of these points is f right or left continuous?
a)       { 2x+1  x<-1
    f(x) { 3x      [-1,1]
          { 2x-1   x>1

Each of the "pieces" is a continuous function (all polynomials are continuous) so the whole question is what happens at -1 and 1.  What are the one sided limits at each of those>

b)      { sqrt(-x)  x<0
   f(x) { 1           [0,1]
         { sqrt(x)   x>1

sqrt(x) is continuous for x non negative so again the only question is what happens at each part.

   You do know that for continuous functions the limit is equal to the value don't you?  That also applies to one sided limits.

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