Applying Leibniz Integral Rule to Constant Limits of Integration

Question

Using the Leibniz integral rule given here, Leibniz, it seems that for any arbitrary multivariable function $f(x,y)$, we have: $$ \int_a^b \frac{\partial}{\partial y} f(x,y) \ dx = \frac{d}{dy} \left(\int_a^b f(x,y) \ dx \right) $$ Is that really always true? I feel like there must be some necessary conditions on $f$ to make that statement valid - like continuity of the function, it's partials, differentiability, etc. - but I am not quite certain. My question is what exactly are these conditions or is it simply always valid?

Answer 1

No it's not always true. Take for example $$f(x,y)=\text{sgn}(x-y).$$