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: 
I'm afraid that from a pre-calculus point of view, there is no consistent definition of the functional relationships that allows to discuss them as real functions (real valued in one real variable).

The big gap to be closed is the notion of an angle. In geometry, an angle is defined by a pair of rays from the same point. These most often occur at corners of triangles. All geometrically similar pairs of rays define the same angle. You can arrange such an equivalent pair of rays to have the first ray horizontal pointing to the right, then the second ray is uniquely characterized by a point on the unit circle (after fixing an origin and a unit length). The coordinates of that point then are the cosine and sine of that (equivalence class of) angle(s).

Then there are some angles that can be constructed and calculated geometrically by constructing regular n-gons and thus dividing the unit circle into n equal parts. n=3,4,6,8,12 are easy, n=5 is possible without too much effort. By bisection one can enclose any other angle in fractions of the full circle, starting from these known fractions, but this goes close to limit arguments, which already is calculus.

One can identify points on the unit circle with rotation matrices or complex numbers, which simplifies the angle arithmetic. Doubling an angle is then squaring the matrix or complex number, bisecting an angle corresponds to a square root. This gives functional equations that in their components are the trigonometric identities. Demanding continuity and differentiability at the zero angle will also give the trigonometric functions as unique solutions. However, this uniqueness is again enforced by calculus arguments.
More precisely, in Analysis, **sin(x)** is defined as the limit of a specific series.

Before, we know that the below series converges for every $x\in\mathbb{R}$:
$$\sum\_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}x^{2n+1}$$
And then we define :
$$\sin(x)=\sum\_{n=0}^\infty \frac{(-1)^n}{(2n+1)!}x^{2n+1}$$
By this definition, $\sin(x)$ has all properties of the elementary definition (ratio of sides in a right-angled triangle).