Hemicontinuity
（重定向自Hemicontinuous）
![An upper hemicontinuous correspondence that is not lower hemicontinuous. It is not lower hemicontinuous because for a sequence of points xm that converges to x, we have a y (y in f(x)) such that no sequence of ym converges to y where each ym is in f(xm).](/Images/godic/202501/17/Upper_hemicontinuous.svg3530.png")
![A lower hemicontinuous correspondence that is not upper hemicontinuous. It is not upper hemicontinuous because the graph (set) is not closed.](/Images/godic/202501/17/Lower_hemicontinuous.svg3530.png")
In mathematics, the notion of the continuity of functions is not immediately extensible to multi-valued mappings or correspondences. The dual concepts of upper hemicontinuity and lower hemicontinuity facilitate such an extension. A correspondence that has both properties is said to be continuous in an analogy to the property of the same name for functions.