Mathematics 12.

In this section we will work exclusively with functions that map a subset of the real numbers to the set of real numbers. The graph of such a function can be illustrated by a set of points in the plane, a curve. There is close connection between the simple properties of the functions and some easily observed properties of the curve.

The graph of the function \latex{ f } intersects the axis \latex{ x } at the function's zeros.

For example the zeros of the function \latex{f:\R\longrightarrow \R, f(x)=x^{2}-1} are \latex{-1} and \latex{1}  as its graph intersects the axis \latex{x} at these two places. (Figure 5)

The zeros of the function \latex{g:\R\longrightarrow \R, g(x)=\sin x} are the values \latex{x=k\Pi ,k\in \Z,} therefore there are infinitely many zeros. The function's graph intersects the axis \latex{ x } at these positions. (Figure 6)

Solving an equation is equivalent to the problem of finding the zeros of a function. For example solving the equation \latex{x^{3}-1=4x^{2}-4x,x\in \R} means that we are looking for the zeros of the function

\latex{f:\R\longrightarrow \R,f(x)=x^{3}-4x^{2}+4x-1.}

Solving the equation \latex{\cos ^{2}x+\cos x\times \sin x=1, x\in \R} can be traced back to the question of finding the zeros of the function

\latex{g:\R\longrightarrow \R,g(x)=\cos ^{2}x+\cos x\times \sin x-1.}

Continuous functions play a significant role in the part of mathematics which studies functions. The precise definition of continuity can only be given using more advanced mathematical tools. Visually: if a function \latex{ f } is continuous at a point \latex{ a }, then changing the value of \latex{ a } only by a little, the image \latex{f(a)} also changes only by a little. The graph of a continuous function can be drawn using a continuous curve. 

An important property of continuous functions is that if \latex{ f } is continuous on the interval \latex{\left[a;b\right]} and the sign of \latex{f(a)} and \latex{f(b)} are different, then \latex{ f } has a zero inside the open interval \latex{\left]a;b\right[.} (Figure 7)

For example the function \latex{f:\R\longrightarrow \R, f(x)=x^{2}} is decreasing on \latex{\left]-\infty;0\right]} while increasing on \latex{\left[0;+\infty\right[.} (Figure 8)

For example the function \latex{f:\R\longrightarrow \R, f(x)=\left[x\right]} is non-decreasing on the whole of its domain. (Figure 9) 

The function \latex{g:\R\longrightarrow \R, g(x)=-\left[x\right]} is non-increasing on its whole domain. (Figure 10) 

It is worth noting that, for example, the constant function \latex{h:\R\longrightarrow \R,h(x)=2}  is both non-increasing and non-decreasing.

For example, the function \latex{f:\R\longrightarrow \R,f(x)=x^{2}} is bounded on the interval \latex{\left[0;2\right]} as any x in it satisfies \latex{0\leq x^{2}\leq 4.}  Obviously \latex{-1\leq x^{2}\leq 10} is also true on \latex{\left[0;2\right]}, therefore \latex{–1} is also a lower bound and \latex{ 10 } is also an upper bound of f on this interval.

For example, the function \latex{f:\left]0;+\infty\right[ \longrightarrow \R,f(x)=\log _{2}x}  is not bounded either from above nor from below on its domain (Figure 11). For any given negative number there exists a small positive number such that for any smaller positive number the value of the function is smaller then the given negative number.

For example \latex{\log _{2}x\lt -1,000,} if

\latex{0\lt x\lt 2^{-1,000}=\frac{1}{2^{1,000} }.}

Similarly \latex{\log _{2}x\gt 1,000,} if

\latex{x\gt 2^{1,000}.}

Obviously, if \latex{ K } is an upper bound, then any number larger than \latex{ K } is also an upper bound. Similarly, if \latex{ k } is a lower bound, then any number smaller than \latex{ k } is also a lower bound.

It is a notable fact that there always exists a smallest among the upper bounds, which is called the supremum, as well as there exists a largest among the lower bounds, which is called the infimum.

The function \latex{f:\R\longrightarrow \R,f(x)=\sin x} is bounded on the whole set of real numbers, its supremum is \latex{1},  its infimum is \latex{–1.} (Figure 6)

The so-called fractional part function, \latex{g:\R\longrightarrow \R, g(x)=x-\left[x\right]} is an interesting example. (Figure 12)

This function is bounded on the whole set of real numbers. Its infimum is \latex{ 0 } which is attained at every integer. Its supremum is \latex{ 1 }, although this value is not attained anywhere by the function. It can be seen that any non-negative value smaller than \latex{ 1 } is attained by the function therefore there cannot exist any upper bound smaller than \latex{ 1 }.

For example the function \latex{f:\R\longrightarrow \R, f(x)=x^{2}-2x+1} has a minimum at  \latex{x=1} and the minimum value is \latex{ 0 }. (Figure 13)

The function \latex{g:\R\longrightarrow \R,g(x)=\cos x} has maximum at the points \latex{x=2k\Pi , k\in \Z} with maximum value being \latex{ 1 }, and it has minimum at the points \latex{x=(2n+1)\times \Pi ,n\in \Z} with minimum value being \latex{–1.} (Figure 14)

It is worth observing that the function \latex{f:\R\longrightarrow \R,f(x)=x-\left[x\right]} does not have maximum as it attains any non-negative value smaller than \latex{ 1 }, but not \latex{ 1 }. (Figure 12)

For example, the function \latex{f:\R\longrightarrow \R,f(x)=2^{x}} s convex on the whole set of real numbers (Figure 15), the function \latex{g:\R^{+}\longrightarrow \R, g(x)=\log _{2}x} is concave on the interval \latex{\left]0;\infty \right[}  (Figure 16). 

We will prove that the function \latex{f:\R^{+}\longrightarrow \R,f(x)=\frac{1}{x^{2} }} is convex on its whole domain. 

It can be shown that the function is continuous on its domain (Figure 17). Thus it suffices to show that for arbitrary \latex{0\lt a\lt b} the following holds:

\latex{\frac{1}{\left(\frac{a+b}{2} \right)^{2} } \lt \frac{\frac{1}{a^{2} }+\frac{1}{b^{2} } }{2}.}

In the denominator of the left-hand side of this inequality there is the square of the arithmetic mean of \latex{ a } and \latex{ b }. Since we know that for \latex{a\neq b} the arithmetic mean is greater than the harmonic mean, that is,

\latex{\frac{2}{\frac{1}{a}+\frac{1}{b} }\lt \frac{a+b}{2},}

it follows that the reciprocal of the square of the arithmetic mean is less than the reciprocal of the square of the harmonic mean,

\latex{\frac{1}{\left(\frac{a+b}{2} \right)^{2} } \lt \left(\frac{\frac{1}{a}+\frac{1}{b} }{2} \right)^{2}.}

Therefore it suffices to show that

\latex{\left(\frac{\frac{1}{a}+\frac{1}{b} }{1} \right)^{2}\lt \frac{\frac{1}{a^{2} }+\frac{1}{b^{2} } }{2}.}

This last inequality is true as the left-hand side is the square of the arithmetic mean of numbers \latex{\frac{1}{a}}  and  \latex{\frac{1}{b}} while the right-hand side is the square of the quadratic mean of the same two numbers and \latex{a\neq b.} 

Thus the proposition is proven.

For example, the function \latex{f:\R\longrightarrow \R,f(x)=x^{2}} is even (Figure 8), the function \latex{g:\R\longrightarrow \R, g(x)=x^{3}} is odd (Figure 18).

The graph of an even function is symmetric to the axis \latex{ y } while the graph of an odd function is centrally symmetric to the origo.

For example the function \latex{f:\R\longrightarrow \R,f(x)=\cos x} is periodic and its fundamental period is \latex{2\Pi}. (Figure 14)

The tangent function is also periodic and its fundamental period is \latex{\Pi}.  (Figure 19)

The following function, the so-called Dirichlet function is a curious example for a periodic function:

\latex{f(x)=\begin{cases} 1, \text{if } x \text{ is rational}, \\ 0, \text{if } x \text{ is irrational} \end{cases}}

It is impossible to draw the graph of this function since it consists of dense sets of points on the axis \latex{ x } and on the line  \latex{y=1} parallel to the axis \latex{ x }.

For this function any positive rational number is a period since if \latex{r\gt 0} is an arbitrary rational number then for any rational \latex{ x } it follows that \latex{x+r} is also rational and thus

\latex{f(x)=f(x+r)=1,}

while for any irrational \latex{ x } it follows that \latex{x + r} is also irrational and thus

\latex{f(x)=f(x+r)=0.}