Show that a closed graph and compact codomain implies continuity.

Asked Apr 25 '19 at 13:19

Active Apr 25 '19 at 13:24

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Let $$f:X\rightarrow Y,$$ where X and Y are metric spaces and Y is compact. If f has a closed graph $$G_f=\{(x,f(x)):x\in X\},$$

then f is continuous.

My first instinct is to use the fact that if $${x_n}\rightarrow x \;\text{ and }\; \{f(x_n)\} \rightarrow y$$ implies x is in X and f(x)=y, then showing the f is continuous at $(x,y)$, and so continuous on X. Having trouble getting direction on this.

edited Apr 25 '19 at 13:24

YuiTo Cheng

4,705

asked Apr 25 '19 at 13:19

CoryTrevor

@YuiToCheng Not sure that this is really a duplicate as the OP is adding the hypothesis that the spaces are metric spaces. This suggests to use a more elementary proof based on sequential compacity. – mathcounterexamples.net Apr 25 '19 at 13:25
@mathcounterexamples.net Then see Analogue of closed graph theorem (That's the sequential argument you are hinting at?) – YuiTo Cheng Apr 25 '19 at 13:26
That is exactly what I'm looking for. Thank you! – CoryTrevor Apr 25 '19 at 13:29
@YuiToCheng Yes, this a real duplicate. – mathcounterexamples.net Apr 25 '19 at 13:32

Show that a closed graph and compact codomain implies continuity.

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