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Discretionary jurisdiction is a circumstance where a court has the power to decide whether to hear a particular case brought before it. Most courts have no such power, and must entertain any case properly filed, so long as the court has subject matter jurisdiction over the questions of law which must be decided, and in personam jurisdiction over the parties to the case.
Typically, the highest court in a state or country will have discretionary jurisdiction. The reason for this is that appellate courts in common law countries have two basic functions: error correction and ensuring the orderly development of case law.
In the first case, the appellate court simply examines the record and determines whether the lower court applied existing law correctly, and reverses and remands (sends the case back) for severe errors. That is, the parties may generally agree on the applicable law, but the appellant will contend that the trial court incorrectly interpreted and applied the existing law.
In the second case, the appellate court rules on novel issues in a case, and under stare decisis, those rulings become new law in themselves. In those cases, the parties disagree vigorously if any existing legal rule even applies to the facts of the case, or the appellant may be deliberately trying to attack an established rule in the hope that the appellate court will overturn a prior decision and establish a new rule, or the question has been ruled upon by multiple intermediate appellate courts and is so perplexing that all the lower courts disagree with each other.
An appellate court with discretionary jurisdiction is able to delegate error correction to lower courts, while it focuses its limited resources on developing case law properly. In the latter situation, the appellate court will focus on truly novel questions or revisiting older legal rules that are now clearly obsolete or unconstitutional.
For example, the United States Supreme Court hears cases by a writ of certiorari, essentially meaning that it calls appellants up to the court only if their case is important enough to merit the resources of the court. The Supreme Court employs a "rule of four", meaning that four justices have to think the case is important enough to hear before the court will grant it review. Many state supreme courts use a similar process to choose which cases they will hear.
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