10. Find all points of discontinuity of f, where f is defined by 

f (x) = \left\{\begin{matrix} x+1 & if \: \: x \geq 1 \\ x^2 +1 & if x \: \: <1 \end{matrix}\right.

Answers (1)

Given function is
f (x) = \left\{\begin{matrix} x+1 & if \: \: x \geq 1 \\ x^2 +1 & if x \: \: <1 \end{matrix}\right.
given function is defined for every real number k 
There are different cases for the given function
case(i)   k > 1
f(k) = k+1\\ \lim_{x\rightarrow k}f(x) = k+1\\ \lim_{x\rightarrow k}f(x) = f(k)
Hence, given function is continuous for each value of k > 1

case(ii)   k < 1
f(k) = k^2 ++1\\ \lim_{x\rightarrow k}f(x) = k^2+1\\ \lim_{x\rightarrow k}f(x) = f(k)
Hence, given function is continuous for each value of k < 1

case(iii)  x = 1

\lim_{x\rightarrow 1^-}f(x) = x^2+1 = 1^2 + 1 = 1 + 1 = 2\\ \lim_{x\rightarrow 1^+}f(x) = x+1 = 1+1 = 2\\ f(1) = 1^2+1 = 2 \\ R.H.L. = L.H.L. = f(1)
 
Hence, at x = 2 given function is continuous
Therefore, no point of discontinuity

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