# Prove the continuity in the composition function.

If $g$ is continuous at $c$

and $f$ is continuous at $g(c)$

Then prove that $(f\circ g)$ is continuous at c.

To prove this I have done something: Given: $$\lim_{x\to c}g(x)=g(c) \tag 1$$ and, $$\lim_{g(x)\to g(c)}f(g(x))=f(g(c)) \tag 2$$ Now,$$\lim_{x\to c}(f\circ g)(x)=\lim_{x\to c}f(g(x))=f(\lim_{x\to c}(g(x)))=f(g(c))=(f\circ g)(c) \tag 3$$

Is the second equality in (3) true ?

If no then please prove it.

And if yes then what is then, what is the requirement of (2)?

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Yes, the second equality is true since we're given $\;f\;$ is continuous at $\;g(c)\;$ and this means that
$$\lim_{x\to g(c)}f(x)=f\left(\lim_{x\to g(c)}x\right)=f(g(c))$$
no matter how $\;x\to g(c)\;$ , and since $\;g(x)\xrightarrow [x\to c]{}g(c)\;$ because we're also given that $\;g\;$ is continuous at $\;c\;$ , we're done.