Explain why continuity along straight lines is not enough to conclude continuity

calculus continuity

Solution 1:

Here's a hint: what if you approach $(0,0)$ along the path $y = x^2$? Note that $f$ is only continuous at $(0,0)$ if every path to $(0,0)$ yields the same limit.

Related

Finding $\lim\limits_{x \to \infty} (\sqrt{9x^2+x} - 3x)$
Entire function having simple zero at the Gaussian integers
Homomorphisms and automorphisms on polynomial rings
Prove $\lim_{n\to\infty}\frac{\int_{-1}^{1}{f(x)(1-x^2)^n}dx}{\int_{-1}^{1}{(1-x^2)^n}dx}=f(0)$
Determine the number of positive integer x where $x\leq 9,999,999$ and the sum of the digits in x equals 31.
How is $ \cos (\alpha / \beta) $ expressed in terms of $\cos \alpha $ and $ \cos \beta $?
Let $f(x) = x^2 \sin (1/x^2),\,x\ne 0,$ and $ f(0)=0.$ Prove $f$ is differentiable on $\mathbb R$
How to find examples of periodic points of the (complex) exponential-function $z \to \exp(z)$?
What does insolvability of the quintic mean exactly?
Cesaro Mean of Sequences - Convergence
Show that $\mathbb{E}(T) = \sum\limits^\infty_{k=1}\mathbb{P}(T \geq k)$ for $T$ nonnegative integer valued and $E[T] < \infty$
How to derive the following identity?