Question

How do you find where `f(x)=1/ln(x)` is continuous and differentiable on which interval(s)?

Answer 1

It's continuous and differentiable on the union `(0,1)\cup (1,infty)`. That is, for all `x>0` except `x=1`.

Explanation:

The function `ln(x)` is continuous and differentiable for all `x>0`. Therefore, `1/(ln(x))` will be continuous and differentiable for all such values of `x` as well, except for those values of `x` where `ln(x)=0`. The only such value of `x` where the logarithm is zero is `x=1`.

Therefore, `1/(ln(x))` is continuous and differentiable on the union `(0,1)\cup (1,infty)`. That is, for all `x>0` except `x=1`.

Moreover, if you are interested, the Quotient Rule gives

`d/dx(1/(ln(x)))=(ln(x) * 0-1 * 1/x)/(ln(x))^2=-1/(x (ln(x))^2)` for `x>0` and `x!=1`.