Ugrásdiszcontinuitás

Ugrásdiszcontinuitás, also known as jump discontinuity, is a type of discontinuity in a function at a particular point. This occurs when the limit of the function as it approaches the point from the left is different from the limit of the function as it approaches the point from the right. Importantly, both the left-hand limit and the right-hand limit must exist as finite numbers for a jump discontinuity to be present.

In simpler terms, if you were to trace the graph of a function with a jump discontinuity, you would see a clear break at that point. You would have to lift your pen from the paper to get to the function's value on the other side of the discontinuity. This is distinct from other types of discontinuities, such as removable discontinuities where a single point can be redefined to make the function continuous, or infinite discontinuities where the function approaches infinity.

Mathematically, a function f(x) has a jump discontinuity at x = c if the following conditions are met: lim x->c- f(x) exists and is finite, lim x->c+ f(x) exists and is finite, and lim x->c- f(x) != lim x->c+ f(x). The value of the function at c, f(c), may or may not exist, or it may be equal to one of the one-sided limits, but it will not bridge the gap between them. Examples can be found in piecewise functions where the definitions change abruptly.