what's the difference between jump and removable discontinuity?

Answer 1

x* is a removable discontinuity if lim x->x*- = lim x->x*+ and is finite. In other words, the limit as x approaches x* from the right is equal to the limit as x approaches x* from the left.

x* is a jump discontinuity if the limit as x approaches x* from the right is different from the limit as x approaches x* from the left.

If f(x) = x^2 when x!=2 and f(x) = 5 when x = 2. Then x=2 is a removable discontinuity. Because the limit as x approaches 2 from the right is 4 and from the left is also 4.

If f(x) = 10 when x>5 and f(x) = 0 when x<=5. Then x=5 is a jump discontinuity becuase the limit as x approaches 5 from the right = 10 while from the left the limit is 0.

Answer 2

Jump Discontinuity

Answer 3

Let's look at it this way. Suppose f(x) = n(x) / d(x) where n and d are continuous functions at x = a. What possible things can happen at the point x = a? 1. If n(a) = anything and d(a) ≠ 0, then f is continuous at x = a and f(a) = n(a) / d(a). 2. If n(a) ≠ 0 and d(a) = 0, then f is undefined at x = a and there is a vertical asymptote at x = a 3. If n(a) = 0 and d(a) = 0. Then f is undefined at x = a. This means that it is also discontinuous at x = a. But if lim(x -->a) f(x) exists, say it equals b, then f has a removable discontinuity at x = a. That is, if WE DEFINE f(a) = b, then f is now continuous at x = a.

Answer 4

In a removeable discontinuity (say at x = k) the limit of the function as x -> k is the same for x approacing from less than k (approachinjg from below) as it is for x approaching from greater than k (approaching from above) and the value of the limit is bounded (that is, not infinite or 'unbounded'). For a jump, the limit is different if x -> k from above rather than below. In addition to this, one or both limits may be unbounded.

Doug

Answer 5

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Answer 6

for removable, you can just plug 1 point in to make it continuous.
for jump, you cant, because it jumps up or down