For what value of $\lambda$ is the function defined by $f$ defined by $f(x) = \left\{ \begin{array} {1 1} \lambda(x^2 - 2x) ,& \quad\text{ if$ x $$\leq 0$}\\ 4x + 1,& \quad \text{if$x$> 0}\\ \end{array} \right.$ What about continuity at $x = 1$?

This question is separated into two parts

Toolbox:
• If $f$ is a real function on a subset of the real numbers and $c$ be a point in the domain of $f$, then $f$ is continuous at $c$ if $\lim\limits_{\large x\to c} f(x) = f(c)$.
At $x=1$
$\lim\limits_{\large x\to 1}f(x)=\lim\limits_{\large x\to 1}(4x+1)$
$\quad\quad\quad\;\;=4+1$
$\quad\quad\quad\;\;=5$
$\Rightarrow f$ is not continuous at $x=0$ for any value of $\lambda$ but continuous at $x=1$ for all values of $\lambda$.