Is this statement about change of variables in limits true?

Question

I was brainstorming on this and on some related thoughts in the past 2-3 days so any help here would be much appreciated. What put me in these thoughts... was one proof (from a textbook) which I read and which was using the implication b) (see below) very casually and as if it is quite natural to assume that this is true. So I realized this needs more rigorous justification.

Is the following statement/theorem true?

Given are the two functions:

$$g(x): (a,b) \rightarrow (c,d) \tag{1}$$

$$f(x): (c,d) \rightarrow \mathbb{R} \tag{2}$$

The function $g$ is continuous, monotonic and takes all values between $c$ and $d$
(it is surjective but possibly not bijective)

Let also: $$\lim\limits_{x \to a} g(x) = c$$

Then:

a) if the limit $\lim\limits_{y \to c} f(y)$ exists and is $L$, then $\lim\limits_{x \to a} f(g(x))$ also exists and is equal to $L$

b) if the limit $\lim\limits_{x \to a} f(g(x))$ exists and is $L$, then $\lim\limits_{y \to c} f(y)$ also exists and is equal to $L$

Note 1: Here the symbols $a,c$ may denote numbers or $-\infty$, and the symbols $b,d$ may denote numbers or $+\infty$

Note 2: By limit exists and equals $L$, it is meant that the limit is either a number, or also an infinity (positive or negative)

I think this theorem is true and it is what justifies when people do simple variable
substitutions in limits (almost mechanically) and write casually that:
$$\lim\limits_{t \to 0+} \phi(1/t) = \lim\limits_{x \to \infty} \phi(x)$$ or say
$$\lim\limits_{t \to \infty} \phi((t-2)^{2}) = \lim\limits_{x \to \infty} \phi(x)$$ or e.g. $$\lim\limits_{t \to 0} \phi(1/t^2) = \lim\limits_{x \to \infty} \phi(x)$$

I think I was able to prove both a) and b).

The part a) is proved easily, it even follows from
Limit of composite functions theorem

But the proof of part b) substantially uses the assumption that $g$
takes all values between $c$ and $d$ (otherwise b) would not hold true, right?).

But my point is that when we do almost mechanically such changes of variables
in limits (from $t$ to $x$ or from $x$ to $y$, etc.)... we don't really think about the Limit of composite functions theorem and check its conditions, right?

Instead we just think of $g$ as bijection and we assume by intuition that changing the variable works OK because of the bijective behavior of $g$.

• Would anyone agree with that?

• And... could someone confirm this theorem is true?

• Also, do we need any restrictions on f in this theorem? I think not.

• Also, can we remove the condition that g is continuous? Or relax the conditions of the theorem in some other way?