All continuous solutions are differentiable.
Let $\mathcal{C}$ be the space of continuous functions over $\mathbb{R}$. Let $f \in \mathcal{C}$ be a function such that $V = \langle f_b : b \in \mathbb{R} \rangle$, the span of $f_b$ as a subspace of $\mathcal{C}$ is two dimenisonal. Let
$$B = \bigg\{ b \in \mathbb{R} : f, f_b \text{ linear dependent on }\mathcal{C} \bigg\}$$
It is clear $B$ is a subgroup of $\mathbb{R}$ for addition. If $0 \in B$ is not an isolated point of $B$, then it is easy to see $B$ is dense in $\mathbb{R}$. Using continuity of $f_b$
w.r.t $b$, it is not hard to verify $f$ is an exponential function and leads to a contradiction that $V$ is one dimensional. As a result, $0$ is an isolated point of $B$.
WOLOG, we will assume $B \cap (-L,L) = \{ 0 \}$ for some $L > 1$ and take $f$ and $f_1$ as a basis of $V$. Let $A : \mathbb{R} \to M_{2}(\mathbb{R})$ be the $2 \times 2$ matrix valued function on
$\mathbb{R}$ defined by:
$$\begin{align}
\begin{pmatrix}f_{b}\\f_{b+1}\end{pmatrix} = A(b) \begin{pmatrix}f\\f_1\end{pmatrix}
\end{align}\tag{*1}$$
Since $f$ and $f_1$ is linear independent, there are $x, y \in \mathbb{R}$ such that
the matrix $\begin{pmatrix}f(x) & f(y)\\f_1(x)&f_1(y)\end{pmatrix}$ is invertible. From $(*1)$, we have
$$\begin{pmatrix}f_b(x)&f_b(y)\\f_{b+1}(x)&f_{b+1}(y)\end{pmatrix} = A(b)
\begin{pmatrix}f(x) & f(y)\\f_1(x)&f_1(y)\end{pmatrix}$$
Since the LHS is a continuous function in $b$, so does $A(b)$. For any $b, c \in \mathbb{R}$, it is clear $A$ satisfies:
$$\begin{cases}
& A(b) A(c) = A(b+c) = A(c)A(b)\\
\text{ and } & A(b) A(-b) = A(-b)A(b) = A(0) = I_2\tag{*2}
\end{cases}$$
This implies $A(b)$ is always invertible.
Since $A(b)$ is continuous in $b$ and $A(0) = I_2$, there exists a $\epsilon > 0$
such that $\| A(b)-I_2 \| < \frac12$ for $b \in (-\epsilon, \epsilon)$. This implies
$$\left\| \int_0^b \left( A(t) - I_2 \right) dt \right\| \le \int_0^b \left\| A(t) - I_2 \right\| dy < \frac{b}{2}\quad\text{ for }b \in (0,\epsilon)$$
As a result,
$\quad\displaystyle\int_0^b A(t) dt = b I_2 + \int_0^b \left(A(t) - I_2\right) dt\quad$
is invertible over $(0,\epsilon)$.
Let $D : (0,\epsilon) \to M_2(\mathbb{R})$ be the function
$\quad\displaystyle D(b) = \left( A(b) - 1 \right)\left( \int_0^b A(t) dt \right)^{-1}$.
For any $b,c \in (0,\epsilon)$, we have
$$\int_0^{b+c} A(t) dt =
\left( \int_0^b + \int_b^{b+c} \right) A(t) dt =
I_2 \left( \int_0^b A(t) dt \right) + A(b) \left( \int_0^c A(t) dt \right)$$
Exchange the role of $b, c$ and subtract, we get
$$
\begin{align}
& \left(A(b) - I_2\right)\left( \int_0^c A(t) dt \right)
-\left(A(c) - I_2\right)\left( \int_0^b A(t) dt \right) = 0\\
\implies & \left(D(b) - D(c)\right) \left( \int_0^b A(t) dt \right) \left( \int_0^c A(t) dt \right) = 0\\
\implies & D(b) = D(c)
\end{align}$$
i.e. $D(b)$ is a constant matrix over $(0,\epsilon)$. We will use the same $D$ to denote this constant matrix. Over $(0,\epsilon)$, $A(b)$ satisfies an integral equation
$$A(b) = I_2 + D \int_0^b A(t) dt$$
and hence $A(b) = e^{D b}$ there. Using $(*2)$, we see this expression is valid for all $b$.
From this, we can conclude $A$ and hence $f$ is differentiable.
