Continuity of implicit function defined by a continuous and strictly monotone function

Question

Function $f:\mathbb R \mathsf x \mathbb R \rightarrow \mathbb R$ is given to be

1. Continuous everywhere.
2. Strictly monotone as per each free variable.
3. Such that the implicit function $g: \mathbb R \rightarrow \mathbb R$ with $f(x,g(x)):=c$ for c a real constant, is well-defined.

Can it be deduced that g is continuous?

Note that $f$ is not given to be differentiable, therefore the Implicit Function Theorem cannot be used.