Showing that function is discontinuous in a point

**bjorno** · September 21st 2010, 02:31 AM

f(x) = sin(pi/x), when x != 0, and 0, when x = 0

Prove that f(x) is discontinuous in x = 0

Is it necessary to use the delta-epsilon-definition, or do you think i could sketch the graph and point to the jump in value?

If I need to use the definition, could someone give me a hint?

**Prove It** · September 21st 2010, 02:42 AM

As $x \to 0, \frac{\pi}{x} \to \infty$ .

Since the sine function oscillates, that means $\sin{\frac{\pi}{x}}$ will oscillate between $-1$ and $1$ quicker and quicker as $x \to 0$ .

Since $\sin{\frac{\pi}{x}}$ will not tend to a single value, that means the limit does not exist and is thus discontinuous.

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