Do all continuous functions have antiderivatives?

Let $I\subset\mathbb R$ be an interval with more than one point. If $f\colon I\longrightarrow\mathbb R$ is a continuous function, the existence of an anti-derivative of $f$ can be proved as follows: take $a\in I$ and define$$\begin{array}{rccc}F\colon&I&\longrightarrow&\mathbb R\\&x&\mapsto&\displaystyle\int_a^xf(t)\,\mathrm dt.\end{array}$$Then $F$ is a primitive of $f$, by the Fundamental Theorem of Calculus.

You seem to find it strange that every continuous has an anti-derivative while not all continuous functions are differentiable, but it's up to you to explain what's strange about it.