Is $\sin (t)/t$ continuous at $t=0$? And also, if a function $f(x)$ is of indeterminate form at $x=a$, can it be continuous if $f(a)$ does not exist? Can a discontinuous function have a local maximum or minimum?
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A function $f$, defined by $f=\frac{\sin x}{x}$ is continuous everywhere except at the point $x=0$. Although the limit of $f$ as $x \to 0$ is $1$, $\frac{0}{0}$ is strictly undefined and so $f$ cannot be continuous there.
However, one can remove the discontinuity by constructing a new function, let's call it $\bar f$ such that $$\bar f(x) \begin{cases} \frac{\sin x}{x}, & \text{if $x\neq0$} \\ 1, & \text{if $x =0$ } \end{cases} $$This new function is continuous at $x=0$ and is, in fact, differentiable at $x=0$.
Mark Viola
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Thanks for replying to the question. One more query, Why cannot we have sinx/x as 1 at x=0 as the limiting value tends to 1? – santhosh gondi Mar 12 '15 at 05:27
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That is the limit. But it is not the value. The value is undefined since $0/0$ has no meaning. Only through the singularity removal process do we end up with a continuous function. Please "Up vote" if this helped. – Mark Viola Mar 12 '15 at 05:31