Problem regarding existence of a function.

Question

Let $f:[0,1]\to \mathbb{R}$ be function satisfying, $$\lim_{x\to a}f(x)= ∞, \forall a\in [0,1].$$ Thus such a function exists?

Answer 1

Lets partition the domain such that we have some $A$ that is a subset of the interval that is the domain of our function.

Suppose $f(x) = p$ when $x\in A$ and $f(x) = q$ when $x\notin A$

If $A$ has finitely many points, then for all $a \in A$

$\lim_\limits{x\to a} = q$

It can't be $p$ because there exists a neighborhood around $a$ such that for all $x$ in the neighborhood (excluding $a$) $f(x) = b$

But if there are infinitely many points, then there exists some $a\in A$ such that every neighborhood of $a$ will have at least one more point in $A$ that is arbitrarily close to $a.$

Either the function is continuous at $a$ and $\lim_\limits{x\to a} = p$ or it is not and the limit does not exist.

Now, that we have that $f(x) = \infty$ for some subset of $[0,1]$ and infinty is not a real number, no such function exists such that $f:[0,1]\to \mathbb R$