If f(x)=x−4|x−4|+a,x<4a+b, x=4x−4|x−4|+b,x>4 Then, f(x) is continuous at x=4 when a2+b2=

# If  Then, f(x) is continuous at

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### Solution:

We have,

$\begin{array}{r}\underset{x\to {4}^{-}}{lim} f\left(x\right)=\underset{x\to {4}^{-}}{lim} \frac{x-4}{|x-4|}+a=\underset{x\to {4}^{-}}{lim} \frac{x-4}{-\left(x-4\right)}+a=a-1\\ \underset{x\to {4}^{+}}{lim} f\left(x\right)=\underset{x\to {4}^{+}}{lim} \frac{x-4}{|x-4|}+b=\underset{x\to {4}^{+}}{lim} \frac{x-4}{x-4}+b=b+1\end{array}$

and,$f\left(4\right)=a+b$

If f(x) is continuous at x = 4, then

We find that f(x) is neither continuous nor differentiable at . At all other points $f\left(x\right)$ is continuous and differentiable. $f\left(|x|\right)$is every where continuous but not differentiable at

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