Question

Prove that the function f(x)= x-3 is continuous but not differentiable

Answer 1

We have to prove that $\left |x-3\right |$ is continuous but not differentiable.

Since, every modulus function is continuous for all real numbers 'x'.

Therefore, f(x)=$\left |x-3\right |$ is continuous at x = 3.

Now, Consider f(x) = 3-x for x<3

Therefore, Left hand limit f '(3)= $lt_{x\rightarrow 3^{-}} \frac{f(x)-f(3)}{x-3}$

$=lt_{h\rightarrow 0}\frac{f(3-h)-0}{3-h-3} = -1$

Now, Consider f(x) = x-3 for x$\geq$3

Therefore, Right hand limit f '(3)= $lt_{x\rightarrow 3^{+}} \frac{f(x)-f(3)}{x-3}$

$=lt_{h\rightarrow 0}\frac{f(3+h)-0}{3+h-3} = 1$

Since, left hand limit is not equal to right hand limit.

Therefore, the given function is not differentiable at x = 3.