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$f(x) = x^2$ if $x\leq 1$,

$f(x) = 1$ if $x>1$.

To show that the above function is continuous at $x=1$, I have tried the following method,

$\epsilon$-$\delta$ definition to show continuity.

What other methods can be used for this example, or in general cases, or in the case of functions mapping from $\mathbb{R}^2$ to $\mathbb{R}^2$.

Example of complex function(analogous to $\mathbb{R}^2 \to \mathbb{R}^2$),

$f(z) = \frac{z^3-2}{z^2+z+1}$ if $|z|\neq 1$

$f(z) = \frac{-1+i\sqrt 3}{2}$ if $|z| = 1$

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See in this example if $$\lim_{x \to 1^-}f(x)=\lim_{x \to 1^+}f(x)=f(1)$$

You can use it also in general cases of piecewise functions to check continuity at an arbitrary point $x_0$ in the domain of the function.

In general see if

$\lim_{x \to x_0}f(x)=f(x_0)$ holds

A function $f(\bar{x})=(f_1(\bar{x})...f_n(\bar{x}))$ from $\mathbb{R}^n$ to $\mathbb{R}^n$ is continuous iff each of its component functions $f_i$ is continuous.

So you have to check the continuity of each component function.

Also a general and handy method is to check the continuity of the function using the sequential characterization of continuity in $\mathbb{R}^n,\forall n \geq 1$(and in metric spaces in general).

See this.

You can use this method also to prove the discontinuity of a function at a given point.

Let me show an example.

Take $f(x)=\begin{cases}\ \sin{\frac{1}{x}} ,& x \neq 0\\ 0 & x=0\\ \end{cases}$

Take the sequence $x_n=\frac{1}{2n \pi+\frac{\pi}{2}} \to 0$.

But $f(x_n)=1 \to 1 \neq 0=f(0)$.

So according to the theorem of sequential continuity,the function is not continuous at zero.

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  • $\begingroup$ how about the case of functions mapping from $\mathbb{R}^2$ to $\mathbb{R}^2$ $\endgroup$ – Little Rookie Sep 6 '17 at 13:39
  • $\begingroup$ @LittleRookie this is a whole different question..when you post a question which receives an answer then after that don't change your original post with adding other questions..You can just ask them in different posts. $\endgroup$ – Marios Gretsas Sep 6 '17 at 13:45
  • $\begingroup$ Yes we have to check if each of its component function is continuous. And that brings back to the original question, which is that besides using the $\epsilon$ - $\delta$ definition, what are the other methods? $\endgroup$ – Little Rookie Sep 21 '17 at 23:22
  • $\begingroup$ @LittleRookie...see my edits $\endgroup$ – Marios Gretsas Sep 21 '17 at 23:44
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    $\begingroup$ +1 for sequential continuity. It is great boon to prove negative results (ie to confirm that a function is discontinuous). $\endgroup$ – Paramanand Singh Sep 22 '17 at 16:45

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