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# 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$> 0}\\ \end{array} \right.$ Continuous at $x = 0$?

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 $x=0$
LHL=$\lim\limits_{\large x\to 0}\lambda(x^2-2x)$
$\quad\;\;\;=0$
RHL=$\lim\limits_{\large x\to 0}(4x+1)$
$\quad\;\;\;=1$
$f(0)$=LHL $\neq$ RHL
$\Rightarrow f$ is not continuous at $x=0$ whatever value of $\lambda\in R$