Function with no maximum on [0,1]

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Give an example of a discontinuous function on $[0,1]$ that has no absolute maximum.

Solution

The example serves to show that, if the continuity hypothesis fails, the conclusion of Weierstrass’s theorem may fail too. Consider $f(x) = \begin{cases} x & 0\le x<1 \\ 0 & x=1 \end{cases}$ defined on the whole of $[0,1]$ (a closed and bounded interval).

• For $x\in[0,1)$ the values $f(x)=x$ get as close to $1$ as we like ($\sup f = 1$), but never reach it.
• At the point $x=1$, where the value $1$ could be attained, the function equals $0$.

Therefore the supremum $1$ is not a maximum: there is no $x\in[0,1]$ with $f(x)=1$. The function is discontinuous at $x=1$ (jump), and it is precisely this discontinuity that prevents the maximum from existing, consistently with the fact that Weierstrass’s theorem requires continuity.

Links

Topics: Continuity
Concepts: Discontinuity · Absolute maxima and minima · Weierstrass’s theorem
Skills: Analysis of limiting cases · Proving
Exercise types: Proof