Consider the family of functions $\{f_c \mid c\in\bf{R}\}$ where $$ \begin{align*} f_c &\colon{\bf R}^2\to{\bf R}\\ &f_c(x,y)=1-\big(x^2+4y^2\big)^c. \end{align*} $$ I intuitively see that this function is not continuous for $c\leq 0$, but how can I prove this with a 'clean' argument involving limits? How can I determine for which $c$ the function $f_c$ is differentiable?
Continuity and differentiability of a elliptic/paraboloid function
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calculus
vector-analysis
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0@Mark Dominus You are both right, I've just fixed it – 2012-03-26
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1Hint: first look at what happens as $x \to 0$ with $y=0$. – 2012-03-26