Let a ∈ R {\displaystyle {}a\in \mathbb {R} } and let f , g , h : R → R {\displaystyle {}f,g,h\colon \mathbb {R} \rightarrow \mathbb {R} } be functions. Suppose that g {\displaystyle {}g} and h {\displaystyle {}h} are continuous in a {\displaystyle {}a} , that g ( a ) = f ( a ) = h ( a ) {\displaystyle {}g(a)=f(a)=h(a)} holds and suppose further that g ( x ) ≤ f ( x ) ≤ h ( x ) {\displaystyle {}g(x)\leq f(x)\leq h(x)} holds for all x ∈ R {\displaystyle {}x\in \mathbb {R} } . Show that also f {\displaystyle {}f} is continuous in a {\displaystyle {}a} .
