Question

Show that ƒ(x) = [x] is neither continuous nor derivable at x=2.

Answer 1

Left hand limit at x = 2

 \(\lim\limits_{x \to 2^-}\) f(x) = \(\lim\limits_{h \to 0}\)f(2-h) = \(\lim\limits_{h \to 0}\)(2-h)= \(\lim\limits_{h \to 0}\) 1 = 1

Right hand limit at x = 2

 \(\lim\limits_{x \to 2^+}\) f(x) = \(\lim\limits_{h \to 0}\)f(2+h) = \(\lim\limits_{h \to 0}\)(2+h)= \(\lim\limits_{h \to 0}\) 2 = 2

As left hand limit ≠ right hand limit 

Therefore, f(x) is not continuous at x = 2 

Lets see the differentiability of f(x): 

LHD at x = 2

As, LHD ≠ RHD

Therefore, 

f(x) is not derivable at x = 2