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How do you tell if a function is continuous?
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A Sometimes, it is easy: A polynomial function is always continuous. Rational functions with non-zero denominators as well as the sine and cosine functions are also continuous. Other continuous functions include root functions, exponential functions, and logarithmic functions.

Intuitively, a function is continuous if you can draw it without lifting your pen from your paper.

Continuous Functions
The function $f(x)$ is continuous at $x=a$ if

$\lim _{\mathrm{x}_{\rightarrow} a-} f(x)=\lim _{\mathrm{x}_{\rightarrow} a^{a+}} f(x)$ i.e. $\lim _{\mathrm{x}_{\rightarrow} a} f(x)$ exists and equals $f(a)$

The function $f$ is said to be continuous on its domain if it is continuous at each point in its domain. If $f$ is not continuous at a particular value $' a$, we say that $f$ is discontinuous at $a$ or that $f$ has a discontinuity at $a$.

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