Properties of Continuous Functions

Introduction 3.3.1  
Now we take a look at a few theorems without proof (the proofs involve ideas of compactness and closedness of sets that would lead us too far away from the essence of these lecture notes). These theorems are intuitively very reasonable and you can convince yourself that they do make sense by trying to graph a couple counterexamples.

Theorem 3.3.2 (Weierstrass)  
If $f : [a,b] \to \hbox {I \hskip -5.2pt {R}}$ is a continuous function, then $f$ attains a minimum and a maximum within the interval $[a,b]$.

Remark 3.3.3  
The picture you should have in mind when thinking about Weierstrass' theorem should be something like:

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The fact that $f$ is continuous excludes functions like $\frac{1}{x}$ considered over the interval $[-1,1]$. In this case, the the maximum is $\infty$ and the minimum $-\infty$. Also note that the theorem no longer works if we consider open intervals. Take e.g. again the function $\frac{1}{x}$, but over the interval $]0,1]$. Over this interval, the function is continuous, and yet, the maximum is infinity.

Theorem 3.3.4 (Bolzano (a.k.a. Mean Value Theorem))  
If $f : [a,b] \to \hbox {I \hskip -5.2pt {R}}$ is a continuous function, then $f$ attains all values between $f(a)$ and $f(b)$.

Remark 3.3.5  
Discontinuous functions will rarely work for Bolzano's theorem as the following picture indicates:

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Clearly, There is no $x$ such that $f(x)=10$. The converse is obviously not true. A function that attains all values between $f(a)$ and $f(b)$ is not necessarily continuous, as the following picture of the $\tan$ function convincingly illustrates:

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Corollary 3.3.6 (Root Existence Theorem)  
If $f$ is continuous over $[a,b]$ and $f(a) \cdot f(b) <0$, meaning that one is positive and the other is negative, then there exists at least one $c$ in $]a,b[$ for which $f(c)=0$.