I am not sure that linearization is applicable to this functional equation, but I will depict another way to solve this. First, lets consider trivial solutions.
$$f^2(x)+1=2f(x+1)$$
$$f(x)=c$$
$$c^2+1=2c$$
$$c^2-2c+1=0$$
$$(c-1)^2=0$$
$$c=1$$
$$f(x)=1$$
I assume that the function is meant to be continuous and defined everywhere. Lets find some restrictions we can put on $f(x)$. The domain must be all real numbers.
$$f(x+1)=\frac{f^2(x)}{2}+\frac{1}{2}$$
$$f(x-1)= \pm \sqrt{2f(x)-1}$$
Both of these operations should be valid if we want $f(x)$ to have a domain of all real numbers. There is no problem scaling a value of $f(x)$ up, but there may be a problem scaling down due to the domain of square roots.
$$2f(x)-1 \geq 0$$
$$2f(x) \geq 1$$
$$f(x) \geq \frac{1}{2}$$
This means that we only need to consider the positive branch of the square root.
$$f(x-1)= \sqrt{2f(x)-1}$$
We will further bound $f(x)$ by analyzing a sequence constructed from the equation.
$$a_{n+1}= \sqrt{2a_n-1}$$
We must ensure that $a_n$ never goes below $\frac{1}{2}$. We will first establish that $a_n$ is a monotonically decreasing sequence.
$$a_n>a_{n+1}$$
$$a_n>\sqrt{2a_n-1}$$
$$a_n^2>2a_n-1$$
$$a_n^2-2a_n+1>0$$
$$(a_n-1)^2>0$$
This is true for all $a_n$ except at $a_n=1$, which is the only solution to $a_n=a_{n+1}$. Since $a_n$ is monotonically decreasing, all $a_n<1$ can be ruled out as they will eventually go below $\frac{1}{2}$. We have tightened the bound on $f(x)$ to:
$$f(x) \geq 1$$
For all $a_n>1$, we must prove that $a_n$ always stays above $1$.
$$\sqrt{2a_n-1}>1$$
$$2a_n-1>1$$
$$2a_n>2$$
$$a_n>1$$
Since $a_n=1$ is the only solution to $a_n=a_{n+1}$, $a_n$ is monotonically decreasing, and $a_n$ stays above $1$ for all $a_n>1$, $a_n$ converges to $1$ for all $a_0>1$. Convergence of this sequence to $1$ and the bound on $f(x)$ means that:
$$\lim_{x \to -\infty} f(x)=1$$
$$\text{From the main identity:}$$
$$f^2(x)+1=2f(x+1)$$
It is clear that the entirety of $f(x)$ can be defined if $f(x)$ is known over a domain interval of length $1\text{.}$ To find the other values $f(x)$ can be shifted using the identity. Let this interval be $x \in [0, 1]\text{.}$ Let $S(x)$ be a continuous function defined on this interval. $S(x)$ can be used contrive other values of $f(x)\text{.}$ For $f(x)$ to be continuous at $x=1$, S(x) must satisfy:
$$S^2(0)+1=2S(1)$$
Generally, any continuous seed function on the $[0, 1]$ interval will allow for computation of a continuous function $f(x)$ given that the seed function satisfies:
$$S(x) \geq 1$$
$$S^2(0)+1=2S(1)$$
Conclusions that can be drawn about the resulting continuous function $f(x)$ include:
$$f^2(x)+1=2f(x+1)$$
$$f(x) \geq 1$$
$$\lim_{x \to -\infty} f(x)=1$$
Additionally, there are a few things that can be said about $f(x)$ as $x$ tends to $\infty\text{.}$
If $S(x)=1$, then $\lim_{x \to \infty}f(x)=1$
If $S(x)>1$, then $\lim_{x \to \infty}f(x)=\infty$
If $S(x)=1$ for some $x$ in the interval but not all $x$, then:
$$\liminf_{x \to \infty}f(x)=1 \quad\quad \limsup_{x \to \infty}f(x)=\infty$$
This occurs because the sequence $b_n$ is monotonically increasing for all $b_n>1$ where:
$$b_{n+1}=\frac{b_n^2}{2}+\frac{1}{2}$$
$$b_{n+1}>b_n$$
$$\frac{b_n^2}{2}+\frac{1}{2}>b_n$$
$$b_n^2+1>2b_n$$
$$b_n^2-2b_n+1>0$$
$$(b_n-1)^2>0$$
Note that using the seed function method to generate $f(x)$ can produce all possible $f(x)$ that are continuous everywhere. The next step would be to determine if any solutions are differentiable everywhere. I will come back to edit this answer if I find any.
