# Continuity

Continuity’s solution is comprehensive and provides the automation, regulatory updates, and customized tracking, monitoring and data organization tools we need for effective compliance management. We look forward to a strong and enduring partnership with Continuity to meet our regulatory requirements.”. Continuity (mathematics), the opposing concept to discreteness; common examples include Continuous probability distribution or random variable in probability and statistics Continuous game, a generalization of games used in game theory Law of Continuity, a heuristic principle of Gottfried Leibniz.

Show Mobile NoticeShow All NotesHide All Notes
You appear to be on a device with a 'narrow' screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

### Section 2-9 : Continuity

1. The graph of (fleft( x right)) is given below. Based on this graph determine where the function is discontinuous. Solution
2. The graph of (fleft( x right)) is given below. Based on this graph determine where the function is discontinuous. Solution

For problems 3 – 7 using only Properties 1 – 9 from the Limit Properties section, one-sided limit properties (if needed) and the definition of continuity determine if the given function is continuous or discontinuous at the indicated points.

1. (displaystyle fleft( x right) = frac{{4x + 5}}{{9 - 3x}})
1. (x = - 1)
2. (x = 0)
3. (x = 3)
Solution
2. (displaystyle gleft( z right) = frac{6}{{{z^2} - 3z - 10}})
1. (z = - 2)
2. (z = 0)
3. (z = 5)
Solution
3. (gleft( x right) = left{ {begin{array}{rl}{2x}&{x < 6}{x - 1}&{x ge 6}end{array}} right.)
1. (x = 4)
2. (x = 6)
Solution
4. (hleft( t right) = left{ {begin{array}{rl}{{t^2}}&{t < - 2}{t + 6}&{t ge - 2}end{array}} right.)
1. (t = - 2)
2. (t = 10)
Solution
5. (gleft( x right) = left{ {begin{array}{rc}{1 - 3x}&{x < - 6}7&{x = - 6}{{x^3}}&{ - 6 < x < 1}1&{x = 1}{2 - x}&{x > 1}end{array}} right.)
1. (x = - 6)
2. (x = 1)
Solution

### Continuity Store

For problems 8 – 12 determine where the given function is discontinuous. 1. (displaystyle fleft( x right) = frac{{{x^2} - 9}}{{3{x^2} + 2x - 8}}) Solution
2. (displaystyle Rleft( t right) = frac{{8t}}{{{t^2} - 9t - 1}}) Solution
3. (displaystyle hleft( z right) = frac{1}{{2 - 4cos left( {3z} right)}}) Solution
4. (displaystyle yleft( x right) = frac{x}{{7 - {{bf{e}}^{2x + 3}}}}) Solution
5. (gleft( x right) = tan left( {2x} right)) Solution

### Continuity Tester

For problems 13 – 15 use the Intermediate Value Theorem to show that the given equation has at least one solution in the indicated interval. Note that you are NOT asked to find the solution only show that at least one must exist in the indicated interval.

### Continuity Math

1. (25 - 8{x^2} - {x^3} = 0) on (left[ { - 2,4} right]) Solution
2. ({w^2} - 4ln left( {5w + 2} right) = 0) on (left[ {0,4} right]) Solution
3. (4t + 10{{bf{e}}^t} - {{bf{e}}^{2t}} = 0) on (left[ {1,3} right]) Solution