If a function is not continuous, does it mean it has no limit? [closed]

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If a function is not continuous, does it mean it has no limit? If it's false can you give me an example? It's not homework I really want to understand

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3 Answers

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No, a function can be discontinuous and have a limit. The limit is precisely the continuation that can make it continuous.

Let $$f(x)=1\text{ for }x=0,\\f(x)=0\text{ for }x\ne0.$$

This function is obviously discontinuous at $x=0$ as it has the limit $0$.

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Consider first the function $I(x)$ which is $1$ on rationals and $0$ on irrationals. If this is multiplied by a function such as $(x-a)$ then the result will be discontinuous everywhere but at $a$ yet the limit as $x \to a$ will exist and be $0$.

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Imagine function $f$ which has the same value $x_n$ on $[n,n+1)$. If $(x_n)$ has a limit, then $\lim_{x \to + \infty}f(x)$ also exists.

More subtle example: let $f(x)=1^x$, $f(0)=0$. $\lim_{x \to 0}f(x)$ exists, even though $f$ isn't continuous in $0$.

Also, for $sign(x)$, $\lim_{x \to +0}f(x)$ exists (as well as $\lim_{x \to -0}f(x)$).

The only bound between continuity and having a limit is that function continuous in point has a limit in that point (which equals its value), not the other way around.

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