Continuity at a point implies continuity in the neighborhood

I claim that if a function ##f:\mathbb{R}\rightarrow\mathbb{R}## is continuous at a point ##a##, then there exists a ##\delta>0## and ##|h|<\frac{\delta}{2}## such that ##f## is also continuous in the ##h##-neighbourhood of ##a##.



Please advice if my proof as follows is correct.



Continuity at...



Continuity at a point implies continuity in the neighborhood

Continuity at a point implies continuity in the neighborhood