Question

check whether every integrable function is differentiable or not ? justify

Answer 1

Solution:

Every integrable function does not need to be differentiable.

A piecewise function (with discontinuity) is integrable but not differentiable.

Example(1): $y = |x|$ is integrable in the domain [-1, 1], like

$$\int y dy =\int_{-1}^1 y dy = \int_{-1}^ 0 (-x) dx + \int _0^1 x dx$$

But this function |x| is not differentiable at x = 0.

Left hand Derivative

$$lim_{x \to 0^-} \frac {f(x) - f{(0)} }{x - 0} =lim_{x \to 0^-} \frac {|x| - 0} {x - 0} = lim_ {x \to 0^-} \frac {-x} {x} \\ = lim_{x \to 0^-} (-1) = - 1$$

Right hand Derivative

$$lim_{x \to 0^+} \frac {f(x) - f{(0)} }{x - 0} = lim_{x \to 0^+} \frac {|x| - 0} {x - 0} = lim_ {x \to 0^+} \frac {x} {x} \\ = lim_{x \to 0^-} (1) = 1$$

Left hand Derivative is not equal to Right hand  Derivative,

So, it is not differentiable.

Example(2): A step function is integrable but not differentiable

Example(3): A Weierstrass function is integrable but not differentiable.