# Logical implication in existence of partial derivatives with non differentiable function

I ask for some help in unknotting this chain of reasonings, in particular in spotting logical errors due to wrong implications and possible mathematical misbeliefs about this problem.

Theorem: If $$f(x, y)$$ has partial derivatives (id est, they exist) at a point $$(x_0, y_0)$$, and they are continuous at $$(x_0, y_0)$$, then $$f$$ is differentiable at $$(x_0, y_0)$$.

This is an IF - THEN, hence the condition partial derivatives exist and they are continuous is sufficient but not necessary.

Indeed I know that $$f$$ can be differentiable even if partial derivatives are not continuous at $$(x_0, y_0)$$.

Another fact is: if $$f$$ is not continuous at $$(x_0, y_0)$$, then $$f$$ is not differntiable at $$(x_0, y_0)$$.

Another fact: the existence of partial derivatives does not guarantees that they are also continuous.

Now, at this point I do ask:

• Suppose $$f$$ is not continuos at $$(x_0, y_0)$$, then if partial derivatives of $$f$$ do both exist at $$(x_0, y_0)$$, then at least one of them is not continuous at $$(x_0, y_0)$$

Is this generally valid? Is this generally wrong? Is this always wrong?

I found this, that might help: function not continuous, partial derivatives exists -> partial derivatives not continuous

My question still remains in the sense of the at least one. Is it valid to say "at least one" instead of "$$f$$ hasn't continuous partial derivatives"? The second statement seems global to me, in the sense that ALL the partial derivatives are discontinuous

• Suppose $$f$$ is not continuous at $$(x_0, y_0)$$, then if partial derivatives of $$f$$ do both exist at $$(x_0, y_0)$$, then at least one of them is not continuous at $$(x_0, y_0).$$

Is this generally valid?

Yes, because:

Theorem: If $$f(x, y)$$ has partial derivatives at $$(x_0, y_0)$$, and they are continuous at $$(x_0, y_0)$$, then $$f$$ is differentiable at $$(x_0, y_0).$$

At $$(x_0, y_0),$$ if $$f$$ is not differentiable, then $$f$$ has no partial derivatives or at least one of its partial derivatives is discontinuous.

Another fact: if $$f$$ is not continuous at $$(x_0, y_0)$$, then $$f$$ is not differentiable at $$(x_0, y_0).$$

Therefore, at $$(x_0, y_0),$$ if $$f$$ is discontinuous, then $$f$$ has no partial derivatives or at least one of its partial derivatives is discontinuous.

Hence, at $$(x_0, y_0),$$ if $$f$$ is discontinuous and has partial derivatives, then at least one of its partial derivatives is discontinuous.