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Convex and lower semi-continuous functions

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These are problems that ask as to which values of a variable a function is a) convex and b) lower semi-continuous. Please see the attached file for the full question.

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A function f is said to be convex if for all x, y and .
Define
We are told to assume f is convex on .
is trivially true when either or (since the rhs of the inequality above is then )
So it remains only to choose in such a way that the convexity condition is satisfied when and is any point in .

So f will be convex provided .

b) A function is lower ...

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