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It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.<br><br>Here are some common dental emergencies:<br>Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.<br><br>At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.<br><br>Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.<br><br>Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.<br><br>Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.<br><br>Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.<br><br>Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.<br><br>In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.<br><br>If you adored this informative article and also you wish to get more info with regards to [http://www.youtube.com/watch?v=90z1mmiwNS8 Washington DC Dentist] generously pay a visit to our own website.
In [[mathematics]], a '''nowhere continuous function''', also called an '''everywhere discontinuous function''', is a [[function (mathematics)|function]] that is not [[continuous function|continuous]] at any point of its [[domain of a function|domain]]. If ''f'' is a function from [[real number]]s to real numbers, then ''f''(''x'') is nowhere continuous if for each point ''x'' there is an ε&nbsp;&gt;&nbsp;0 such that for each δ&nbsp;&gt;&nbsp;0 we can find a point ''y'' such that |''x''&nbsp;&minus;&nbsp;''y''|&nbsp;&lt;&nbsp;δ and |''f''(''x'')&nbsp;&minus;&nbsp;''f''(''y'')|&nbsp;≥&nbsp;ε. Therefore, no matter how close we get to any fixed point, there are even closer points at which the function takes not-nearby values.
 
More general definitions of this kind of function can be obtained, by replacing the [[absolute value]] by the distance function in a [[metric space]], or by using the definition of continuity in a [[topological space]].
 
==Dirichlet function==
One example of such a function is the [[indicator function]] of the [[rational number]]s, also known as the '''Dirichlet function''', named after German mathematician [[Peter Gustav Lejeune Dirichlet]].<ref>Lejeune Dirichlet, P. G. (1829) "Sur la convergence des séries trigonométriques qui servent à répresenter une fonction arbitraire entre des limites donées" [On the convergence of trigonometric series which serve to represent an arbitrary function between given limits], ''Journal für reine und angewandte Mathematik'' [Journal for pure and applied mathematics (also known as ''Crelle's Journal'')], vol. 4, pages 157 - 169.</ref> This function is written ''I''<sub>'''Q'''</sub> and has [[domain of a function|domain]] and [[codomain]] both equal to the [[real number]]s. ''I''<sub>'''Q'''</sub>(''x'') equals 1 if ''x'' is a [[rational number]] and 0 if ''x'' is not rational. If we look at this function in the vicinity of some number ''y'', there are two cases:
*If ''y'' is rational, then ''f''(''y'')&nbsp;=&nbsp;1. To show the function is not continuous at ''y'', we need to find an ε such that no matter how small we choose δ, there will be points ''z'' within δ of ''y'' such that ''f''(''z'') is not within ε of ''f''(''y'')&nbsp;=&nbsp;1.  In fact, 1/2 is such an ε.  Because the [[irrational number]]s are [[dense set|dense]] in the reals, no matter what δ we choose we can always find an irrational ''z'' within δ of ''y'', and ''f''(''z'')&nbsp;=&nbsp;0 is at least 1/2 away from 1.  
*If ''y'' is irrational, then ''f''(''y'')&nbsp;=&nbsp;0. Again, we can take ε&nbsp;=&nbsp;1/2, and this time, because the rational numbers are dense in the reals, we can pick ''z'' to be a rational number as close to ''y'' as is required. Again, ''f''(''z'')&nbsp;=&nbsp;1 is more than 1/2 away from ''f''(''y'')&nbsp;=&nbsp;0.
In plainer terms, between any two irrationals, there is a rational, and vice versa.
 
The ''Dirichlet function'' can be constructed as the double pointwise limit of a sequence of continuous functions, as follows:
 
:<math>f(x)=\lim_{k\to\infty}\left(\lim_{j\to\infty}\left(\cos(k!\pi x)^{2j}\right)\right)</math>
 
for integer ''j'' and ''k''.
 
This shows that the ''Dirichlet function'' is a [[Baire function|Baire class]] 2 function. It cannot be a Baire class 1 function because a Baire class 1 function can only be discontinuous on a [[meagre set]].<ref>{{cite book
  | last = Dunham
  | first = William
  | title = The Calculus Gallery
  | publisher = Princeton University Press
  | date = 2005
  | pages = 197
  | isbn = 0-691-09565-5 }}</ref>
 
In general, if ''E'' is any subset of a [[topological space]] ''X'' such that both ''E'' and the complement of ''E'' are dense in ''X'', then the real-valued function which takes the value 1 on ''E'' and 0 on the complement of ''E'' will be nowhere continuous. Functions of this type were originally investigated by [[Peter Gustav Lejeune Dirichlet]].
 
==Hyperreal characterisation==
A real function ''f'' is nowhere continuous if its natural [[Hyperreal number|hyperreal]] extension has the property that every ''x'' is infinitely close to a ''y'' such that the difference ''f(x)-f(y)'' is appreciable (i.e., not [[infinitesimal]]).
 
==See also==
*[[Thomae%27s function]] (also known as the popcorn function) &mdash; a function that is continuous at all irrational numbers and discontinuous at all rational numbers.
 
==References==
<references />
 
==External links==
* {{springer|title=Dirichlet-function|id=p/d032860}}
* [http://mathworld.wolfram.com/DirichletFunction.html Dirichlet Function &mdash; from MathWorld]
* [http://demonstrations.wolfram.com/TheModifiedDirichletFunction/ The Modified Dirichlet Function] by George Beck, [[The Wolfram Demonstrations Project]].
 
[[Category:Topology]]
[[Category:Mathematical analysis]]
[[Category:Types of functions]]

Revision as of 00:38, 18 February 2014

It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.

Here are some common dental emergencies:
Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.

At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.

Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.

Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.

Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.

Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.

Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.

In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.

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