Module 4 : Local / Global Maximum / Minimum and  Curve Sketching
Lecture 11 : Convex / Concave Functions [Section 11.2]
11.2.3 Proposition:
 

Let . Then is convex if and only if for
                                                                       
 
  Proof:
 

Follows from the definition by putting
Geometrically, it should be obvious that if we want to draw graph of a function on an interval on which it is concave upward/ downward, it must be continuous. For example if there is a break in the graph at then the function cannot be concave up/ down:

 
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