Is there a word for this semi-continuity property?

Question

Let $f \colon \mathbb{R} \to \mathbb{R}$ be a function. Recall that $f$ is upper-semicontinuous if $$ \text{for any } x \in \mathbb{R}, \quad f(x) \geq \inf_{\varepsilon > 0} \sup_{|y-x| < \varepsilon} f(y). \tag{$\dagger$}$$ Note that if $g \colon \mathbb{R} \to \mathbb{R}$ is continuous and $f$ is obtained from $g$ by changing the value of $g$ at a single point, that is $$ f(x) = \begin{cases} g(x) &\text{ if } x \neq x_0, \\ y_0 &\text{ if } x = x_0, \end{cases} $$ where $y_0 > g(x_0)$, then $f$ is upper-semicontinuous. This strikes me as a slightly undesirable behaviour, since an upper-semicontinuous function cannot be reconstructed from its values at $\mathbb{R} \setminus \{x_0\}$. Hence, I'm curious if there is an established name for the following stronger property: $$ \text{for any } x \in \mathbb{R}, \quad f(x) = \inf_{\varepsilon > 0} \sup_{0<|y-x| < \varepsilon} f(y). \tag{$\ddagger$}$$ Any references would be highly appreciated. Note that I'm primarily interested in terminology that is already established, the task of coming up with a new name would be inherently opinion-based.