The "standard" definition of upper semicontinuity at a point $x_0$ in a metric space seems to be
$\limsup_{x\to x_{0}} f(x)\le f(x_0)$.
However, why is it this weaker condition instead of
$\limsup_{x\to x_{0}} f(x)= f(x_0)$ ?
For example, with right continuity we have
$\limsup_{\{x\ge x_0\}} f(x)=f(x_0)$.
Compare this situation with the four Dini derivatives (lower left, upper left, upper right, lower right). A function is right differentiable at a point if and only if its upper right and lower right Dini derivatives exist and coincide.
Analogously, a function is right continuous at $x_0$ if and only if its lower right and upper right limits exist, coincide, and agree with the value of the function at $x_0$.
So why would we want to define upper semicontinuity as "upper left and upper right limits exist, do not have to coincide, and are less than or equal to the value of f at $x_0$?".
Wouldn't the definition "upper left and upper right limits exist, coincide, and agree with the value of f at $x_0$" be either more natural or more useful?
Among the useful properties of semicontinuity is that the infimum of any collection of upper semicontinuous functions is upper semicontinuous. In fact, on a metric space this characterizes the upper semicontinuous functions: every u.s.c. function is the infimum of a sequence of continuous functions. This would not be the case with your proposed new property. For example, the function $$ f(x) = \cases{1 & if $x = 0$\cr 0 & otherwise\cr}$$ is the infimum of the continuous functions $\exp(-n x^2)$ for integers $n$, so we want this to be upper semicontinuous.