# Does a graph need to be continuous at a point to have a local extremum at said point?

Say for example we have a composite function with this graph.

and we take the shown interval $[a,b]$ , can we say that this graph has a local minimum at the point $x=c$?

• No, it is an infimum. – Yves Daoust Jul 17 '17 at 17:34
• No, it has a local maximum at $c$ – but it is not a maximum on $(a,b)$. – Bernard Jul 17 '17 at 17:35
• @Yves Daoust: an infimum??? – Bernard Jul 17 '17 at 17:37
• Oh so what you (Bernard) are saying is that since all points near x=c have a lower value than f(c) then it counts as a local maximum ? I can kinda understand that – Dahen Jul 17 '17 at 17:40
• An infimum? I don't think I've studied that yet, can you give me a short explanation on what it is? – Dahen Jul 17 '17 at 17:41