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Consider the function defined by :

$f(p) = p \ln(p) + (1-p) \ln(1-p)$

Show that this extends to a continuous function on $[0,1]$. Does it obtain a global maximum and minimum?

I am struggling with the first part of the question. I know how to remove discontinuity for algebraic functions but for this one I have no clue. In particular, I tried to manipulate the function in order to be able to find a value at $p=0$ and $p =1$ but it never converges to a finite limit.

Any help would be greatly appreciated !

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    I tried to put everything in the log function but it didn't help. When I replace p by 0 or 1 it goes to infinite. – 2017-01-28
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    That's true. I suspect it is the point of the problem. What sort of class or work is this for? It will give a better idea of what to do. – 2017-01-28
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    This is for Optimization of functions in one variable. – 2017-01-28
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    So, it is already continuous on $(0,1)$. Did you perhaps mean $[0,1]$, or is the exercise to prove the continuity property? – 2017-01-28
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    Yes, exactly. The problem asks to prove it is continuous on [0,1]. – 2017-01-28
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    In fact it is continuous at $1$ as well, unless I'm greatly mistaken. Plugging in $1$ gives zero. – 2017-01-28
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    There was a typo in my post. I just corrected it. It is 1-p in the log function. – 2017-01-28
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    This is giving me a time, alright. I'll look back at it in the morning. – 2017-01-28
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    Any new idea to solve the problem ? – 2017-01-28
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    Sadly, no. I thought it would be a simple limit problem, but it is not, at least not from my scratch work. Sorry! :( – 2017-01-29
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    I solved it. It was not that bad actually. At p = 0, problem lies in the first part p ln p. So you find the limit at 0 by composition p ln p = - ln (1/p) / (1/p) and you find 0. Do the same for p = 1. – 2017-01-29
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    Post the solution as an answer. People love that! – 2017-01-29

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I solved it. It was not that bad actually. At p = 0, problem lies in the first part p ln p. So you find the limit at 0 by composition p ln p = - ln (1/p) / (1/p) and you find 0. Do the same for p = 1.