Let $ E $ be a $ mathbb R $-Space Banach and $ (T (t)) _ {t ge0} $ to be a semi-group on $ E $.

Definition:

- Yes $ x in E $then $ (T (t)) _ {t ge0} $ is called
strongly continues to $ x $Yes Yes $$[0infty)Toe\\;tmapstoT(t)xtag1$$[0infty)Toe\\;tmapstoT(t)xtag1$$[0infty)toE;;;;tmapstoT(t)xtag1$$[0infty)toE;;;;tmapstoT(t)xtag1$$ is continuous- $ (T (t)) _ {t ge0} $ is called
locally delineated$ ^ 1 $ Yes Yes $$ exists t> 0: sup_ {s in[0t)}left|T(s)right|_{mathfrakL(E)}<inftytag2$$[0t)}left|T(s)right|_{mathfrakL(E)}<inftytag2$$[0:t)}left|T(s)right|_{mathfrakL(E)}

**Question**: Let $ x in E $ with $$ left | T (h) x-x right | _E xrightarrow {h to0 +} 0 tag4. $$ Are able to show that $ (T (t)) _ {t ge0} $ is strongly continuous to $ x $?

It's easy to see $ (1) $ is just-continuous. In addition, I am able to show that $ (1) $ is left coninue, **as long as I guess $ (T (t)) _ {t ge0} $ is bounded locally**.

Can we abandon the hypothesis of local delimitation and still conclude continuity on the left?

My idea is: by $ (4) $, $$ exists t> 0: forall h in[0t):left|T(h)xright|_E<1+left|xright|_ETag5$$[0t):left|T(h)xright|_E<1+left|xright|_ETag5$$[0t):left|T(h)xright|_E<1+left|xright|_Etag5$$[0t):left|T(h)xright|_E<1+left|xright|_Etag5$$ and perhaps we can show in a certain way the local delimitation by the principle of uniform delimitation.

$ ^ 1 $ By the semigroup property, $ (1) $ implies that $$ sup_ {s in I} left | T (s) right | _ { mathfrak L (E)} < infty $$ for each bounded interval $ I subseteq[0infty)$[0infty)$[0infty)$[0infty)$.