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: 
The proof is based simply on similar triangles. If a right-angled triangle has an angle $\theta$ then the other two angles are $90^{\circ}$ and $(90-\theta)^{\circ}$. If two triangles have the same angles then they are similar.

My picture shows two similar triangles: $\triangle OAB$ and $\triangle OA'B'$. 

Since $\theta = \angle AOB$ then, by definition 
$$\sin\theta = \frac{\|AB\|}{\|OB\|}$$

Since $\theta = \angle A'OB'$ then, by definition
$$\sin\theta = \frac{\|A'B'\|}{\|OB'\|}$$

We can show that $\sin \theta$ has a single, unique value if we can show that the two ratios agree.

Let $T$ be the linear transformation given by an enlargement, centre $O$, with scale factor $\lambda$, such that $T(A) = A'$ and $T(B) = B'$. We have $\|A'B'\|=\lambda\|AB\|$ and $\|OB'\| = \lambda\|OB\|$, hence
$$\frac{\|A'B'\|}{\|OB'\|} = \frac{\lambda\|AB\|}{\lambda\|OB\|}=\frac{\|AB\|}{\|OB\|}$$

This shows that the ratio of the opposite side to the hypotenuse is the same for any two similar, right-angled triangles with angle $\theta$. That means that $\sin\theta$ is uniquely, well-defined.

(N.B. Similarity allows rotation and reflection as well as enlargement. However, rotations and reflections preserve lengths and so preserve ratios of lengths.)

![enter image description here](https://i.stack.imgur.com/CXmZ5.png)
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.