Example

At the point $x = 0$ the function $x^{1/3}$ (the inverse function of $x^3$, a homeomorphism of $\mathbf{R}$ onto itself) has a derivative, equal to $+\infty$; at $x = 0$ the function $|x|^{1/3}$ has right derivative $+\infty$ and left derivative $-\infty$.

The formulae for the derivative of a sum, of a product of differentiable real functions, and for the inverse of a differentiable function (props. 1, 3 and 4), as well as for the derivative of a (real-valued) composition of functions (prop. 5) remain valid when the derivatives that occur are infinite, so long as all the expressions that occur in these formulae make sense (Gen Top., IV, p. 345–346). In fact, if in prop. 6 one supposes that $f$ is strictly increasing (resp. strictly decreasing) and continuous on I, and if $f'(x_0) = 0$, then the inverse function $g$ has a derivative equal to $+\infty$ (resp. $-\infty$) at the point $y_0 = f(x_0)$; if $f'(x_0) = +\infty$ (resp. $-\infty$), then $g$ has derivative 0. There are similar results for right and left derivatives, which we leave to the care of the reader.

Let C be the graph or representing curve of a finite real function $f$, the subset of the plane $\mathbf{R}^2$ formed by the points $(x, f(x))$ where $x$ runs through the set where $f$ is defined. If the function $f$ has a finite right derivative at a point $x_0 \in I$, then the half-line with origin at the point $M_{x_0} = (x_0, f(x_0))$ of C, and direction numbers $(1, f'_d(x_0))$ is called the right half-tangent to the curve C at the point $M_{x_0}$; when $f'_d(x_0) = +\infty$ (resp. $f'_d(x_0) = -\infty$) we use the same terminology for the half-line with origin $M_{x_0}$ and direction numbers $(0, 1)$ (resp. $(0, -1)$). In the same way one defines the left half-tangent at $M_{x_0}$ when $f'_g(x_0)$ exists. With these definitions one can verify quickly that the angle which the right (resp. left) half-tangent makes with the abscissa is the limit of the angle made by this axis with the half-line originating at $M_{x_0}$ and passing through the point $M_1 = (x, f(x))$ of C, as $x$ tends to $x_0$ while remaining $> x_0$ (resp. $< x_0$).

One can also say that the right (resp. left) half-tangent is the limit of the half-line originating at $M_{x_0}$ passing through $M_1$, on endowing the set of half-lines having the same origin with the quotient space topology $C^*/\mathbf{R}_+^*$ (Gen. Top., VIII, p. 107).

If the two half-tangents exist at a point $M_{x_0}$ of C, they are in opposite directions only when $f$ has a derivative (finite or not) at the point $x_0$ (assumed interior to I); they are identical only when $f'_d(x_0)$ and $f'_g(x_0)$ are infinite and of opposite sign. In these two cases we say that the line containing these two half-tangents is the tangent to C at the point $M_{x_0}$.

When the tangent at $M_{x_0}$ exists it is the limit of the line passing through $M_{x_0}$ and $M_1$ as $x$ tends to $x_0$ remaining $\neq x_0$, the topology on the set of lines which pass through a given fixed point being that of the quotient space $C^*/\mathbf{R}^*$ (Gen. Top., VIII, p. 114).

The concepts of tangent and half-tangent to a graph are particular cases of general concepts which will be defined in the part of this Series devoted to differentiable varieties.