Question: I got this question:

Let $f$ be the integrable function defined on the interval $[0,2]$ by the rule:

$f(x)=
\begin{cases} 
4x^3 & \text{if $0 \leq x \leq 1$} \\
x^2+2 & \text{if $1<x \leq 2$} \\
\end{cases}$

Does there exist real numbers $c1$ and $c2$ such that the function $F$ defined below is an antiderivative for $f$ (i.e. $F'=f$)?

$F(x)=
\begin{cases} 
x^4 + c\_{1} & \text{if $0 \leq x \leq 1$} \\
\frac{x^3}{3}+2x+c\_{2} & \text{if $1<x \leq 2$} \\
\end{cases}$

I got stuck since the function $f$ got step/jump discontinuity at $x=1$.
And I don't know how to proceed from there.

Thanks.

Answer: 
Usually one begins studying the $\sin$ function in connection with right triangles. Here the notion of angle is not questioned, so I shall stick with this intuitive concept. It follows that we have $\sin \theta$ defined for $0\leq\theta\leq{\pi\over2}$.

As a next step one considers polar angles in the $(x,y)$-plane, measured counterclockwise from the positive $x$-axis, and one identifies these angles with the length of the corresponding arc on the unit circle $S^1$. When $0\leq\theta\leq{\pi\over2}$ then $\sin\theta$ is the $y$-coordinate of the point where the second leg of the angle in question intersects $S^1$. It is then *natural to define* $\sin \theta$ for all $\theta\in[0,2\pi]$ in this way, and a further step leads to the definition of $\sin\theta$ for all $\theta\in{\mathbb R}$: For arbitrary $\theta>0$ spool a thread of length $\theta$ counterclockwise around $S^1$ with initial point at $(1,0)$. The $y$-coordinate of the endpoint is then *defined* to be $\sin\theta$. It is then intuitively obvious that the function $\sin$ is periodic.

This sounds all very simple and natural. The difficulties begin when we try to make the above thought process mathematically precise. In the first place we need a precise notion of angle: axiomatic euclidean geometry allows comparison and addition of angles, but does not provide an identification of angles with real numbers, which is inherent in the concept of the $\sin$ function.
It follows from the definition of $\sin(\theta)$ as a ratio of length of $\theta$'s opposite side to hypotenuse ([Wikipedia](http://en.wikipedia.org/wiki/Sine)).

As long as hypotenuse length is not equal to zero (otherwise the triangle degenerates into line or point), the division is a function which yields only one associated value.