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Yet another common false belief is :

"For non constant periodic functions $f: \mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$ with smallest positive periods $p_1 , p_2$ respectively, the sum $f+g$ is periodic if and only if $\frac{p_1}{p_2}$ is rational."

One side of the statement above is true (the latter implies the former, but the former does not necessarily imply the latter. (But it's true for continuous functions.)

As an exercise one may like to prove the following : (Source : Miklos Schweitzer competition)

Given any two positive real numbers $p_1,p_2$, there exists functions $f_1 : \mathbb{R} \to \mathbb{R}$ with smallest positive period $=p_1$, and $f_2 : \mathbb{R} \to \mathbb{R}$ with smallest positive period $=p_2$, such that $f_1+f_2$ is also periodic.

One more interesting false belief is :

"If $f: \mathbb{R} \to \mathbb{R}$ is a continuous function taking unequal values at points $x_1,x_2 \in \mathbb{R}$ with $x_1 < x_2$ then there is some sub-interval of $(x_1,x_2)$ on which $f$ is either strictly increasing or strictly decreasing."

This is wrong. Among the most familiar counterexamples is the Devil's staircase.

I believe this false belief is often due to the habit of seeing continuous functions wearing the glasses provided by the Intermediate Value Property these functions have.

Addendum : Some nice examples from wikipdia, here : https://en.wikipedia.org/wiki/Normally_distributed_and_uncorrelated_does_not_imply_independent

Yet another common false belief is :

"For non constant periodic functions $f: \mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$ with smallest positive periods $p_1 , p_2$ respectively, the sum $f+g$ is periodic if and only if $\frac{p_1}{p_2}$ is rational."

One side of the statement above is true (the latter implies the former, but the former does not necessarily imply the latter. (But it's true for continuous functions.)

As an exercise one may like to prove the following : (Source : Miklos Schweitzer competition)

Given any two positive real numbers $p_1,p_2$, there exists functions $f_1 : \mathbb{R} \to \mathbb{R}$ with smallest positive period $=p_1$, and $f_2 : \mathbb{R} \to \mathbb{R}$ with smallest positive period $=p_2$, such that $f_1+f_2$ is also periodic.

One more interesting false belief is :

"If $f: \mathbb{R} \to \mathbb{R}$ is a continuous function taking unequal values at points $x_1,x_2 \in \mathbb{R}$ with $x_1 < x_2$ then there is some sub-interval of $(x_1,x_2)$ on which $f$ is either strictly increasing or strictly decreasing."

This is wrong. Among the most familiar counterexamples is the Devil's staircase.

I believe this false belief is often due to the habit of seeing continuous functions wearing the glasses provided by the Intermediate Value Property these functions have.

Addendum : Some nice examples from Wikipedia, here : https://en.wikipedia.org/wiki/Normally_distributed_and_uncorrelated_does_not_imply_independent

Yet another common false belief is :

"For non constant periodic functions $f: \mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$ with smallest positive periods $p_1 , p_2$ respectively, the sum $f+g$ is periodic if and only if $\frac{p_1}{p_2}$ is rational."

One side of the statement above is true (the latter implies the former, but the former does not necessarily imply the latter. (But it's true for continuous functions.)

As an exercise one may like to prove the following : (Source : Miklos Schweitzer competition)

Given any two positive real numbers $p_1,p_2$, there exists functions $f_1 : \mathbb{R} \to \mathbb{R}$ with smallest positive period $=p_1$, and $f_2 : \mathbb{R} \to \mathbb{R}$ with smallest positive period $=p_2$, such that $f_1+f_2$ is also periodic.

One more interesting false belief is :

"If $f: \mathbb{R} \to \mathbb{R}$ is a continuous function taking unequal values at points $x_1,x_2 \in \mathbb{R}$ with $x_1 < x_2$ then there is some sub-interval of $(x_1,x_2)$ on which $f$ is either strictly increasing or strictly decreasing."

This is wrong. Among the most familiar counterexamples is the Devil's staircase.

I believe this false belief is often due to the habit of seeing continuous functions wearing the glasses provided by the Intermediate Value Property these functions have.

Yet another common false belief is :

"For non constant periodic functions $f: \mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$ with smallest positive periods $p_1 , p_2$ respectively, the sum $f+g$ is periodic if and only if $\frac{p_1}{p_2}$ is rational."

One side of the statement above is true (the latter implies the former, but the former does not necessarily imply the latter. (But it's true for continuous functions.)

As an exercise one may like to prove the following : (Source : Miklos Schweitzer competition)

Given any two positive real numbers $p_1,p_2$, there exists functions $f_1 : \mathbb{R} \to \mathbb{R}$ with smallest positive period $=p_1$, and $f_2 : \mathbb{R} \to \mathbb{R}$ with smallest positive period $=p_2$, such that $f_1+f_2$ is also periodic.

One more interesting false belief is :

"If $f: \mathbb{R} \to \mathbb{R}$ is a continuous function taking unequal values at points $x_1,x_2 \in \mathbb{R}$ with $x_1 < x_2$ then there is some sub-interval of $(x_1,x_2)$ on which $f$ is either strictly increasing or strictly decreasing."

This is wrong. Among the most familiar counterexamples is the Devil's staircase.

I believe this false belief is often due to the habit of seeing continuous functions wearing the glasses provided by the Intermediate Value Property these functions have.

Addendum : Some nice examples from wikipdia, here : https://en.wikipedia.org/wiki/Normally_distributed_and_uncorrelated_does_not_imply_independent

Yet another common false belief is :

"For non constant periodic functions $f: \mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$ with smallest positive periods $p_1 , p_2$ respectively, the sum $f+g$ is periodic if and only if $\frac{p_1}{p_2}$ is rational."

One side of the statement above is true (the latter implies the former, but the former does not necessarily imply the latter. (But it's true for continuous functions.)

As an exercise one may like to prove the following : (Source : Miklos Schweitzer competition)

Given any two positive real numbers $p_1,p_2$, there exists functions $f_1 : \mathbb{R} \to \mathbb{R}$ with smallest positive period $=p_1$, and $f_2 : \mathbb{R} \to \mathbb{R}$ with smallest positive period $=p_2$, such that $f_1+f_2$ is also periodic.

Yet another common false belief is :

"For non constant periodic functions $f: \mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$ with smallest positive periods $p_1 , p_2$ respectively, the sum $f+g$ is periodic if and only if $\frac{p_1}{p_2}$ is rational."

One side of the statement above is true (the latter implies the former, but the former does not necessarily imply the latter. (But it's true for continuous functions.)

As an exercise one may like to prove the following : (Source : Miklos Schweitzer competition)

Given any two positive real numbers $p_1,p_2$, there exists functions $f_1 : \mathbb{R} \to \mathbb{R}$ with smallest positive period $=p_1$, and $f_2 : \mathbb{R} \to \mathbb{R}$ with smallest positive period $=p_2$, such that $f_1+f_2$ is also periodic.

One more interesting false belief is :

"If $f: \mathbb{R} \to \mathbb{R}$ is a continuous function taking unequal values at points $x_1,x_2 \in \mathbb{R}$ with $x_1 < x_2$ then there is some sub-interval of $(x_1,x_2)$ on which $f$ is either strictly increasing or strictly decreasing."

This is wrong. Among the most familiar counterexamples is the Devil's staircase.

I believe this false belief is often due to the habit of seeing continuous functions wearing the glasses provided by the Intermediate Value Property these functions have.