$f(x)=\int_{0}^{x} \{5+|1-y|\}dy$ if $x<2$ $$f(x)=\begin{cases} \int_{0}^{x}\{5+|1-y|\}dy & \text{if }x<2 \\
5x+2 &\text{if }x\ge 2
\end{cases}$$
I need to check continuity and differentiability at $x=2$
$$\lim_{x\uparrow 2}f(x)=\int_{0}^{2} \{5+|1-y|\}dy=10+\int_{0}^{1}(1-y)dy-\int_{1}^{2}(1-y)dy=11$$
and $$\lim_{x\downarrow 2}f(x)=10=f(2)$$
so $f$ is not continuous and differentiable at $2$, am I right? Thank you for confirmation.
 A: $\newcommand{\+}{^{\dagger}}%
 \newcommand{\angles}[1]{\left\langle #1 \right\rangle}%
 \newcommand{\braces}[1]{\left\lbrace #1 \right\rbrace}%
 \newcommand{\bracks}[1]{\left\lbrack #1 \right\rbrack}%
 \newcommand{\ceil}[1]{\,\left\lceil #1 \right\rceil\,}%
 \newcommand{\dd}{{\rm d}}%
 \newcommand{\ds}[1]{\displaystyle{#1}}%
 \newcommand{\equalby}[1]{{#1 \atop {= \atop \vphantom{\huge A}}}}%
 \newcommand{\expo}[1]{\,{\rm e}^{#1}\,}%
 \newcommand{\fermi}{\,{\rm f}}%
 \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,}%
 \newcommand{\half}{{1 \over 2}}%
 \newcommand{\ic}{{\rm i}}%
 \newcommand{\iff}{\Longleftrightarrow}
 \newcommand{\imp}{\Longrightarrow}%
 \newcommand{\isdiv}{\,\left.\right\vert\,}%
 \newcommand{\ket}[1]{\left\vert #1\right\rangle}%
 \newcommand{\ol}[1]{\overline{#1}}%
 \newcommand{\pars}[1]{\left( #1 \right)}%
 \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
 \newcommand{\pp}{{\cal P}}%
 \newcommand{\root}[2][]{\,\sqrt[#1]{\,#2\,}\,}%
 \newcommand{\sech}{\,{\rm sech}}%
 \newcommand{\sgn}{\,{\rm sgn}}%
 \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}}
 \newcommand{\ul}[1]{\underline{#1}}%
 \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert}$
\begin{align}
\color{#0000ff}{\large\left.\fermi\pars{x}\right\vert_{x\ <\ 2}} & = \int_{0}^{x}\pars{1 + \verts{y - 1}}\,\dd y
=
x\pars{1 + \verts{x - 1}} - \int_{0}^{x}y\bracks{\sgn\pars{y - 1}}\,\dd y
\\[3mm]&=
x\pars{1 + \verts{x - 1}} - \half\,x^{2}\sgn\pars{x - 1} +
\int_{0}^{x}\half\,y^{2}\bracks{2\delta\pars{y - 1}}\,\dd y
\\[3mm]&=\color{#c00000}{\large%
x\pars{1 + \verts{x - 1}} - \half\,x^{2}\sgn\pars{x - 1} + \Theta\pars{x - 1}}
\end{align}
where $\ds{\sgn\pars{x}}$, $\ds{\delta\pars{x}}$ and $\ds{\Theta\pars{x}}$ are the
Sign,
Dirac Delta and
Heaviside Step Functions, respectively.

$$\color{#00f}{\large%
\fermi\pars{x}=\left\lbrace%
\begin{array}{lcl}
-\,\half\,x\pars{x - 4} & \mbox{if} & x \leq 1
\\[2mm]
\phantom{-\,}\half\,x^{2} + 1 & \mbox{if} & x > 1
\end{array}\right.}
$$

