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Apr
10
comment Let $E \subset ℝ^n$ open and $f:E→ℝ^m$. Then is $f$ cont. diffb. on $E$ $⇔$ all the partial derivatives $D_jf_i$ exists on $E$ and are cont. on $E$.
@PeterTamaroff: Oh, that makes sense.
Apr
10
comment Let $E \subset ℝ^n$ open and $f:E→ℝ^m$. Then is $f$ cont. diffb. on $E$ $⇔$ all the partial derivatives $D_jf_i$ exists on $E$ and are cont. on $E$.
Wait, isn't this the definition of continuously differentiable?
Apr
10
comment Prove that $f$ is continuous, $f'$ is bounded…
I'm guessing your $x = 0$ and $x \neq 0$ conditions are mixed up.
Apr
10
comment For which numbers $c$ is there a number $x$ such that $f(cx)=f(x)$?
Which exercise is this? I think there's something I'm not understanding.
Apr
10
comment For which numbers $c$ is there a number $x$ such that $f(cx)=f(x)$?
What's $f$ here?
Apr
10
comment Why translation of vectors doesn't preserve the cosinus of the angle they form?
How can you add a scalar to a vector?
Apr
9
comment Gradient of scalar potential
By the square of a vector, do you mean the square of its length?
Apr
9
revised Mathematical Induction: how do we know what applies to one thing also applies to another?
added 1857 characters in body
Apr
9
comment Mathematical Induction: how do we know what applies to one thing also applies to another?
@Hal: We don't have two examples. $P(2)$ fails, because $p_{2+1} - p_2 = p_3 - p_2 = 5- 3 = 2 \ne 1$. The good thing about induction is that we don't have to check all the natural numbers, because we can't: There are infinitely many of them! We don't prove a statement by checking enough special cases. Give me a while and I'll add a bit of clarification to my answer.
Apr
9
comment Solution of $ax=a^x$
In general, you can't find a simple expression for the solution.
Apr
9
answered Mathematical Induction: how do we know what applies to one thing also applies to another?
Apr
8
comment Nonzero derivative implies function is strictly increasing or decreasing on some interval
I'm trying to prove this without assuming the derivative is continuous, but first I gotta ask: is it actually true?
Apr
7
comment Nonzero derivative implies function is strictly increasing or decreasing on some interval
Wouldn't the fact that derivatives satisfy the intermediate value property suffice?
Apr
7
comment Jacobian of the change of variables
Maybe it would help to show what you did and what you got.
Apr
7
comment Calculus inverse function solving for $f^{-1}$
@user71317: $f$ is either always positive or always negative. $f(1) = 2$ says that it's always positive.
Apr
7
comment Interval of definition of the solutions of $\dot x=e^x\sin x$
Isn't every continuous function locally Lipschitz?
Apr
6
answered Help with $\arcsin(x)$ derivative and differentials.
Apr
5
answered Suppose that $f$ is a real valued function such that its second derivative is discontinuous.Can you give some example?
Apr
5
comment Does $a^n \mid b^n$ imply $a\mid b$?
@awllower: you're right that it's a duplicate, but I think this one has better answers.
Apr
5
accepted Does $a^n \mid b^n$ imply $a\mid b$?