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.
If you want to verify this intuitively for yourself, then this answer could prove to be helpful. But remember, *this is not a proof*.

First try to visualize what $\sin\theta$ looks like. If you need some help, this is what it looks like.

[![Alt text](https://i.stack.imgur.com/vLA0e.gif)](https://i.stack.imgur.com/vLA0e.gif)

Now remember that if $\sin\theta$ is a function, then for a certain $\theta$, there can be only *one* value of $\sin\theta$.

Now pick a random point on the $x-$axis. This is going to be your value for $\theta$. Now a draw a line perpendicular to the $x-$axis through that point. Now *is it possible for that line to intersect the sine curve at more than one point?* Try it out for different values of $x$. No, it is completely impossible for that line to intersect that curve at more than one point.

This means that for a certain $\theta$, there is only one associated value of $\sin\theta$.

$\sin\theta$ and $\cos\theta$ are pretty much the same. Notice that $\cos\theta=\sin(\theta +\frac{\pi}{2})$. So if you move the sine curve by $\frac{\pi}{2}$ to the right, you're going to get the curve for $\cos\theta$. And $\sin\theta$ and $\cos\theta$ *look* exactly the same. So you can use the same argument.

Finally, I repeat that this is not a proof. But I hope this helps you see it intuitively.

The figure is taken from [here](http://jwilson.coe.uga.edu/emat6680/dunbar/Assignment1/sine_curves_KD.html).